Zakir Naik, the Indian Muslim preacher, is one of the Islamic world's best-known 'televangelists'. His Peace TV station, broadcasting in Dubai, Saudi Arabia, and Mumbai, has become influential across the Indian subcontinent and in the South Asian Muslim communities abroad. His latest ten-day 'International Islamic Conference', in Hindi and Urdu, has just finished showing in Mumbai over ten evenings this November.
This year Naik's publicity promised the participation of Shaikh Saud Ash-Shuraim, a Wahhabi imam from the Grand Mosque (Haram) in Mecca. It was the second year that Ash-Shuraim lead the Friday prayer at Naik's Mumbai 'peace conference'. Yet his brand of peace has got him into plenty of hot water.
He is banned from Britain and Canada for inciting young Muslims to 'practice terror'.
He has become notorious for his Wahhabism and his past sympathy for Deobandism, the form of Islam that spawned the Afghan Taleban. In addition, he has indulged in incitement against other practices and interpretations in Islam, most notably Sufi devotions and Shiism, and against non-Muslim believers. He flamboyantly advertises conversions from Hinduism, Christianity, and other religions to Islam 'inspired' by his appeal.
Opposition to Naik is localized but severe. He has been censured by leading Indian Muslim clerics, notably Maulana Mufti Abdul Irfan Qadvi, the qazi of Lucknow, who in 2008 condemned his support for Osama Bin Laden.
Qadvi called for Peace TV to be banned; that the Indian government investigate its financing; and that Naik be treated as alien to the Muslim community.
Nevertheless, coincidentally with his attack on Sufis, whose more mystical tradition dominates Indian Islam, Naik offended Shia Muslims by praising the murderer of Imam Husayn, the grandson of Prophet Muhammad.
Last year, Darul Uloom, the authoritative seminary at Deoband, distanced itself from Naik in a fatwa condemning his adoption of 'non-Islamic dress'. His adoption of the Western suit and tie in which he appears commonly, purports to convince his audience of his commitment to dialogue and peace.
But the Deobandis are in agreement with Naik in attempting to prevent celebrations of the birthday (Milad) of Prophet Muhammad. Early in November, the Deobandi clerics of Darul Uloom issued a fatwa forbidding observance of Milad, even though it is an official holiday in every Muslim country in the world, and is recognized as well by countries with large Muslim minorities such as Fiji, Guyana, and Tanzania. The one exception is Saudi Arabia.
The fatwa was rejected by traditional Sunni and Sufi clerics but is indicative of the strengthening of Deobandi-Wahhabi ties, evidenced by the visit in April this year of chief imam of the Grand Mosque, Abdur-Rahman Al-Sudais.
Proclamation of fatwas and teaching of Quranic interpretation without the benefit of religious training is a long-standing characteristic of Wahhabis and their imitators.
Peace TV has featured a number of well-known Wahhabi figures, including the late Ahmed Deedat, who lived in South Africa, the American Siraj Wahhaj, and the Canadian Jamal Badawi.
Naik's 'peace' rhetoric may be perceived as conforming to the new phase of apparent conciliation between Pakistan and India. But for several years, Wahhabis, Deobandis and jihadist followers of the neo-fundamentalist Abu'l Ala Mawdudi have felt encouraged to penetrate the traditional institutions of Islam in India such as the All-India Muslim Personal Law Board.
Until recently, the board was headed by a member of the Barelvi sect of Sunni Muslims. Barelvis, probably a majority among Indian and Pakistani Muslims, and numerically dominant in Britain, are theological conservatives but political realists, who recognise the Muslim obligation to obey the laws of countries in which Muslims are not a majority.
The AIMPLB has become a battleground between the Barelvis and the Deobandis. This is particularly troubling to Indian Muslims as Islam in India has always been Sufi-flavored.
Like Britain and the US, India appears to be making the mistake of thinking that accommodating extremist Islamists will facilitate peace. But it will not.
Pakistani Hindus are fleeing to India as radicalism increases and the political order breaks down under the rule of Islamabad, which protects terrorists and fosters hatred of India from childhood on. Hindus have been murdered in Sindh. This reality outweighs the emptiness of Zakir Naik's 'peace' vocabulary. His Wahhabi agitation is dangerous for Indian Muslims as well as non-Muslims.


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SUPBall: the Sexiest Sport Known to Man
March 18, 2014 by Arianwen Morris 18 Comments
By chance, I stumbled upon a new craze this weekend that's taking Sydney by storm.
Unless you've been hiding under a rock the past few years, or at least venturing nowhere near the ocean, you will be aware of the growing popularity of paddle boarding.
It was only a matter of time before someone turned it into a team sport, and where better to get involved than on a sunny beach on the East Coast of Australia?
Recently, Manly Kayak Centre, situated by the wharf in Manly on the Northern Beaches of Sydney, devised the game SUPBall, otherwise known as Stand Up Paddle Ball. As the name might suggest, you can also rent kayaks here, which is a great way to see some of Sydney's more secluded beaches and get some exercise while you're at it.
While their website describes the new sport as a mix of netball, water polo, lacrosse and rugby, what I witnessed this weekend – as I dangled my bare feet over the edge of the wharf pier and focused my camera lens on some of the most beautiful bodies I've ever seen – was more reminiscent of semi-naked wrestling with the occasional throw of a ball.
SPEEDOS AND MUSCLES
Everywhere I looked were incredibly well-built men in luminous speedos grappling at each other and splashing around in the water. It didn't seem to matter whether the ball was anywhere near them. They appeared to be tackling each other just for the sake of it. Having said that, if I'd been out there among the writhing testosterone-fueled hunks of muscle, I would probably have taken every opportunity to cop a feel too…
The basic premise of the game is simple. Two teams of five go up against each other and aim to score as many goals as possible by throwing a small 'match' ball at their opponent's 'SUPBall goal buoy' and making contact. The game is split into two halves, each lasting 20 minutes.
I was fairly convinced there were no more rules, as chaos more or less ensued for the full game, but it turns out there are a few…
The rules of SUPBall:
› Once a player is in possession of the ball they must stop paddling.
› A player who is not on a board is deemed to be out of the game until they get back on a board. They must not interfere with the ball or other players while they are in the water (well, at least this explains all the tackling).
› Once a goal is scored, the team that scored the goal must retreat around their defensive 'goal buoy' in order to be considered onside. Until they have done this, they must not interfere with the ball or other players (of course, there had to be an offside rule…).
What I love about this game is that pretty much anything goes. You can tip your opponent into the water, steal the ball, and even get on their board with them if it helps you remove them from the game for a few seconds.
And then, well, the views aren't bad either.
If you fancy a game, you'll need to head down to the Manly Kayak Centre on Manly Wharf and register before the start of the next competition season. This year's competition has been running since last October, with matches on Wednesday and Sunday afternoons between 5 pm and 7 pm.
This season, the Manly Kayak Centre encouraged teams of seven to 14 players to sign up. Included in the entry fee was board rental for all games, $15/h board hire at all other times for training purposes and an initial two-hour training session. They were also asked to design their own costumes. Good job boys.
SUPBall is a fantastic way to stay fit, get some sunshine and fresh air and have a lot of fun with your friends.
Alternatively, why not pop down to the Wharf Bar, find yourself a spot on the pier overlooking the action and enjoy ogling the bronzed bodies, while making the excuse that you're following the game. If there was ever a spectator sport for the ladies, I'm certain I've found it.
As luck would have it, Manly is one of the first stops on the CONTIKI BEACHES AND RAINFORESTS tour, which is an amazing way to take in all of the best adventures between Sydney and Cairns.
FOLLOW THE EAST COAST WITH CONTIKI…
AND HAVE NO REGRETS!
Katherine Belarmino says
That is hilarious. Are speedos a requirement of the game?
Katherine Belarmino recently posted..Shanghai by Sidecar, an Insiders Experience
No, not at all. And women and kids play it too. This post is a little misleading, as it's definitely not just for hunks in trunks. I couldn't resist sharing my Sunday guilty pleasure with you ladies though. All the teams were asked to come up with team kits, and these guys decided that speedos were the most appropriate atire. Who am I to argue…? ;)
Hcat says
I suppose in theory the're not a requirement of water polo either, but …
Tamara Lowe via Facebook says
Nice! ;-)
TammyOnTheMove says
Very nice photos Ari! You made my day! :-p
TammyOnTheMove recently posted..Smoking gods, love potions, and llama foetuses in La Paz
Yes, I know the feeling! Those boys made my day last Sunday! :)
visit this page says
This is a fabulous page. Do you play in any sports? My dad love to play all types of darts games at pubs with steel darts. If you are interested in darts please visit this page
I'm more into cardio sports. I used to train in taekwondo and now I love to run. I'm dangerous with darts! Thanks for the heads up though.
Jessica of HolaYessica says
This looks like so much fun! What a great idea. And, as you so wisely pointed out, the views aren't too shabby either. :)
Jessica of HolaYessica recently posted..Learning Spanish with Fun Cat Expressions
It was definitely fun to watch. The only thing that concerned me about playing it was that it's a contact sport but swimwear doesn't offer much in the way of protection. These guys were throwing themselves at each other. You'd definitely get a few bruises!
AlainSojourner says
Oh dear! I think I'd be enjoying just by looking at these guys! of course, with my margaritas. LOL….
AlainSojourner recently posted..Angkor Wat is the Center of the Universe
It's really not a bad way to spend a Sunday afternoon! :)
I would be challenged enough to stand upright on a paddle board on DRY LAND! It does sound like it would be fun to watch especially with the "anything goes…" rules. But…umm…could you do a followup post of this with all females in bikinis? hee hee :) Btw….your new header is kick ass awesome, Arianwen! I LOVE it! :)
Mike recently posted..Top 10 Tom Cruise Movies, Show Me The Money!
I keep meaning to try paddle boarding myself, but each time I have it planned, the weather turns. I don't think I'll be very good at it either! There were some girls there that day actually, and they were equally as fit. I'll see what I can do…!
The Travel Sisters says
This looks like it would be fun to watch!
The Travel Sisters recently posted..Hugging a Panda in Chengdu, China!
Oh yes! It certainly is. Sadly, the SUPBall season just ended. I need to find a new sport to perv over… Any suggestions welcome!

redpajama_set_name: RedPajamaCommonCrawl


One versus Eight. David versus Goliath. Any way you want to frame it the Boston Bruins are heavy favorites over the Detroit Red Wings. But it's tough to call the Red Wings- a team who's now made the playoffs 23 consecutive seasons, with proven veterans- a true underdog, and Coach Claude Julien agrees.
One thing the Wings had for them heading into Friday's game was a couple hits to the Bruins overall depth.
With Chris Kelly out, no longer centering one of the league's best third lines and providing a reliable penalty kill presence, Justin Florek got thrown into the fire. Daniel Paille was knocked out the second-to-last regular season game after a hit to the head, forcing him to miss Friday's contest as he tries to recover. Jordan Caron took his spot.
Kevan Miller and Matt Bartkowski- who had both vehemently proved their top-sixth worth for Game 1 play- both sat out due to injury. Corey Potter and deadline acquisition Andrej Meszaros filled in.
Most of the talk surrounding the Bruins has been about their top two lines, but the team's typically been at it's best when their depth supports their defensive system and provides an offensive punch.
With Kelly, Paille, Bartkowski and Miller all out- they were tested from the get-go.
The replacements faired well in the first period of play. Caron ignited the crowd with an early hit, and created a chance with a centering pass later in the first. Potter also had a noticeably successful period, logging the fourth most ice-time on the Bruins back end, including effective penalty kill time. Florek was also effective on the penalty kill.
Although the Wings outshot the Bruins 11-9 and had the only man-advantage, the game was scoreless after 20.
The second period told a different tale. The Wings were all over the Bruins. It's not confirmed that this "flu bug" is still taking its toll on some players, but the Bruins certainly played like it in the second. Legless, lifeless, the team was down 21-9 in shots attempted and playing with fire. It was as a great start for the Wings, but the Bruins still had time to find their game. We thought.
While the team showed signs of life early in the third period, and even garnered some momentum with a power play opportunity, the sustained attack just wasn't there. Detroit was getting to pucks first, eliminating any chance Boston had at their cycle game, halting any offensive momentum the thought they had.
The game was over just moments after the Bruins best opportunity. After Jarome Iginla fed Milan Lucic with the perfect deflection-oriented pass, it looked like the Bruins would go up 1-0; but the puck trickled through the crease, and it was Pavel Datsyuk with this magician-like hands, corralling the puck at the other end of the rink and ripping a wrist shot past Rask giving the Wings a 1-0 lead and ultimately, the first game of the series.
While the gut-punch will be tough for most for most Bruins fans to swallow- the team's been here before; and quite often at their best in these exact situations as well.
It's unfortunate, and you would think with lessons learned paired with veteran leadership the team has gotten past the point of needing that adversity, that extra emotion to bring out their will to win and the best of their play, but that remains to be seen.


Historically pushed toward softball, Baseball for All keeps young girls in the game.
This week, we're sharing stories from Jennifer Gonnerman, Evan Allen, Britni de la Cretaz, Jen Banbury, and Gordon Edgar.
"Play like a man, look like a lady." At Narratively, Britni de la Cretaz looks at the history of lesbianism in early pro women's baseball and at the beautiful love stories that the movie "A League of Their Own" chose to ignore.


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Q: Delete all contents of directories with given name leaving directory intact Considering the answers for this question about deleting folders by name and this question about stopping searches at the first positive in a path tree, how can I delete all contents of a directory, including sub-directories and files, while leaving the directory intact and stopping at the first match in the path tree?
In other words I might have:
./path/to/foo/again/foo

With the ./path/to/foo folder containing a bar.txt file, and I would be looking to obtain:
./path/to/foo

Containing no files or directories.

A: Note the second link ("about stopping searches at the first positive in a path tree") is not really relevant because a solution that ignores it may empty ./path/to/foo/again/foo and then empty ./path/to/foo; or it may empty ./path/to/foo first and not get to ./path/to/foo/again/foo at all (because this path does not exist now). In any case the goal will be reached.
Anything under (some) foo matches -path '*/foo/*', but the ./path/to/foo itself does not match. find does not add trailing slashes to paths it creates and examines. An exception would be something like find /path/to/foo/ … where /path/to/foo/ will be processed with a trailing slash and it will match -path '*/foo/*'. You may consider -path '*/foo/?*', but if you start from /path/to/foo// (which is a valid pathname) then it will match.
So maybe -path and -name?
find . -path '*/foo/*' ! -name foo

Will it give you a good list of pathnames to delete? Note ./path/to/foo/again/foo will not be on it. The list will contain ./path/to/foo/again, so if you delete again properly then ./path/to/foo/again/foo will be deleted anyway. Not bad, until you realize there will be no entry on the list that allows you to delete ./path/to/foo/foo/foo (if such path exists).
Another problem with -path '*/foo/*' is when you supply a starting point in a form of /a/foo/starting/point. In such case every path considered by find will match. In effect you may inadvertently remove the entire point, while your intention was to empty /a/foo/starting/point/whatever/foo/.
If you can cd to the starting point then this should be safe:
cd /whatever/the/starting/point
# check if cd succeeded (the check can be automated with cd … && find … )
find . -path '*/foo/?*' -delete

(? is most likely not needed because the specification explicitly states there should be no trailing slash in the portion of the pathname relative to what we supplied as the starting path; but I don't know if all implementations of find in a wild strictly obey; so ? is just in case).
Even if foo appears in the path to the starting/point, it wont matter. In find what -path tests is the pathname build by the tool, not the actual absolute pathname. When find starts from . then all pathnames it builds will start with . and -path will test them as strings.
Note -delete is not portable. A portable solution may be one of these:
find . -depth -path '*/foo/?*' -exec rm -r {}  +
find .        -path '*/foo/?*' -exec rm -r {} \;

The first one works like -delete (which implies -depth): it processes a directory only after all its content has been processed. rm will get pathnames one by one in the right order and it will not really need to work recursively because any directory it tries to remove should already be empty (still -r is needed so it can remove directories). Multiple pathnames will be supplied to a single rm thanks to +, still there will probably be more than one rm because you cannot run rm with arbitrarily many arguments. find -exec … + knows this and can execute as many commands (rm processes in our case) as it needs to process arbitrarily many entries and not hit the limit (for comparison: xargs also knows this).
In many cases the most optimal portable solution would be to run one or few rm -r processes that remove non-directories and directories directly under ./path/to/foo/, so find does not even need to go deeper. This is what the second command does. Note there is ; (escaped from the shell) instead of +. We don't want + because it might happen find decides the content of ./path/to/foo/ is not enough to run rm with many arguments yet, it goes deeper to add more arguments, finds the content of ./path/to/foo/again/foo/, only then it starts rm. So rm -r gets ./path/to/foo/again and (possibly among others) ./path/to/foo/again/foo/whatever, it removes the former (recursively), then it complains the latter does not exist. Therefore when without -depth, it's better to run one rm per pathname, so -exec … \;.
But even then files that are no longer there will probably trigger warnings; not from rm but from find. It's because find learns about the content of a directory before processing the directory. It does this even if it's going to process the directory first (i.e. even if there is no -depth). After rm removes ./path/to/foo/again find will still try to process ./path/to/foo/again/whatever. Any solution with (explicit or implied) -depth is free from this problem.
Use the first portable form, the one with -depth; or better the form with -delete, if only your find supports -delete.

Side note: if you want to empty a specific directory then cd to it and:
# check if in the right directory
find . ! -name . -delete

With common implementations of find you don't even need ! -name . because they don't delete . and it's by design. The command simplifies to:
find . -delete

redpajama_set_name: RedPajamaStackExchange


Scintillating sweetheart neckline mermaid gown designed by Alyce ParisGlam up for the upscale evening party with this gorgeous mermaid gown designed by Alyce Paris. The figure-flattering strapless bodice has a romantic sweetheart neckline with that glides slowly into the midriff and later the skirt. The mermaid skirt flares from the knees into a multilayered ruffled bottom for a stunning finish. Pair the dress with pearl jewelry for a fetching look.


Kills all the processes that have matching environment variables.
In this method, the method is given a "model environment variables", which is a list of environment variables and their values that are characteristic to the launched process. The implementation is expected to find processes in the system that inherit these environment variables, and kill them all. This is suitable for locating daemon processes that cannot be tracked by the regular ancestor/descendant relationship.


Hey Readers! Pay attention to download the TNPSC Technical Assistant Hall Ticket 2019 which will be soon released by the Tamil Nadu Public Service Commission. www.tnpsc.gov.in Sr & Jr Technical Assistant Admit Card 2019 will be uploaded at its authorized portal before 10-15 days of the written exam. Individual candidates who have registered for TNPSC Recruitment 2019 can download their respective Tamil Nadu PSC Junior Technical Assistant Admit Card 2019 by using the direct link attached below. The download link will be activated when the authority announced the TN Public Service Commission TA Call Letter 2019 officially. Read out this entire article to know the complete information about the TNPSC Hall Ticket 2019. Bookmark this page to have the latest updates and active links of Tamil Nadu TA Admit Card 2019.
Very soon the TNPSC authority conducting the Written Examination, Interview rounds for the candidates who have successfully submitted their online applications for Senior Technical Assistant and Junior Technical Assistant Posts. So candidates get ready to take up the written exam with the proper exam preparation. If you want the best study material for better practice refers to our All India Jobs website. While attending the written exam candidates should collect their TNPSC Technical Assistant Hall Ticket 2019 @tnpsc.gov.in. The officials have offered the online mode to download the TN Public Service Commission TA Call Letter 2019. Login to the authorized portal of TNPSC and provide the login credentials of registartion number and password. Go through the below sections of this page to get the brief information about the TNPSC Written Exam.
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What Proposed Tax Cuts Mean for Cannabis: Only a Bill?
In a prior post, we discussed how federal income tax applies to cannabis businesses and the huge burden imposed by Section 280E. You may have read about the major tax cuts being proposed in Congress recently which are meant to spur economic and industrial growth across America. Surely Congress would encourage the super-fast growing, multi-billion-dollar cannabis industry by reforming Section 280E, one of the biggest drags on growth, right?
Well, not exactly. As of this writing, the House of Representatives and the Senate have each proposed a tax cut bill, and neither one would make any change to Section 280E. Some members of Congress introduced propose amendments, but nothing made it into the bills.
All is not lost, however. The tax cuts that would benefit non-cannabis businesses would also benefit cannabis businesses, reducing some of the tax bite which could make it easier to conduct business. But there is still a tricky interplay with Section 280E and the tax reform proposals, and understanding how that could affect your business could be key to your business' future growth.
Below is an explanation of the major changes that could impact cannabis businesses. Note that there are, in some cases, significant differences between the House and Senate versions of the bill. Furthermore, the bills are subject to change as they continue to move through the legislative process. (In fact, it is possible that no tax bill is passed at all).
Both the House and Senate bills would lower business tax rates from 35% to 20%, effective in either 2018 (House bill) or 2019 (Senate bill). Pass-through businesses (such as partnerships, LLCs, Subchapter S corporations) would be capped at a top rate of either 25% (House) or 31.8% (Senate, after taking into account the 17.4% deduction). Cannabis businesses, which have higher-than-average effective tax rates because of 280E, would see those rates tumble from the current 60-70% to ~45-55% for corporations and ~40-60% for pass-throughs.
Both the House and Senate bills would allow a 100% immediate write-off of certain capital expenditures ("bonus depreciation"). This would be a boon to non-plant-touching/non-280E cannabis businesses that have fixed asset expenditures. In contrast, plant-touching cannabis businesses can only recover depreciation through "cost of goods sold" because of 280E. It is not clear whether the relevant inventory tax regulations require depreciation to be computed in accordance with GAAP principles (which may not allow bonus depreciation).
Both bills would significantly expand the number of taxpayers that can use the cash method of accounting in calculating taxable income. Under the cash method, a taxpayer would include as income money actually or constructively received and would deduct money actually paid. Currently, a C-corporation or a partnership with C-corporation owners cannot use the cash method if its average annual gross receipts exceed $5 million. A farming C-corporation cannot use that method if its average annual gross receipts exceeds $1 million. The House bill would raise these limits for all companies to $25 million; and the Senate Bill would peg $15 million as the new threshold.
Expanding use of the cash method is meant to be a business simplification proposal. It is far easier for small businesses to track cash, rather than tracking payables and receivables. Plant-touching cannabis businesses can use this simplification, with an important caveat: they would still need to keep careful track of inventory cost of goods sold, since that is the only permitted business deduction under 280E.
Some other positive proposals: both bills would repeal the corporate and individual alternative minimum tax (AMT). This is a secondary tax regime that sometimes unexpectedly imposes additional tax on taxpayers. Also, both bills would retain the research and development (R&D) tax credit, so ancillary non-plant touching businesses in the industry would continue to benefit if they research or develop a new product or process unrelated to the controlled substance. 280E would continue to disallow the R&D credit for plant-touching cannabis businesses, however.
Both bills would negatively affect the cannabis industry by repealing the domestic production activities deduction (Section 199). Section 199 currently allows a deduction equal to 9% of qualified profit (from the sale of items produced, grown, extracted, etc. – in the U.S.). The deduction is currently available to ancillary businesses that do not traffic in controlled substances (280E prevents plant-touching business from taking this deduction).
So, while neither the proposed House nor Senate bills would impact Section 280E, both could significantly impact cannabis businesses and the growth of the industry at large. We expect positive changes to result from business tax rate reductions, capital expenditure incentives, repealing the AMT, and simplicity from increased cash method users. Repealing Section 199 is a definite step back, though.


text:
\section{Introduction}\label{sec:introduction}
Choice functions provide an elegant unifying mathematical framework for studying set-valued choice: when presented with a set of options, they generally return a subset of them.
If this subset is a singleton, it provides a unique optimal choice or decision.
But if the answer contains multiple options, these are incomparable and no decision is made between them.
Such set-valued choices are a typical feature of decision criteria based on imprecise-probabilistic uncertainy models, which aim to make reliable decisions in the face of severe uncertainty.
Maximality and E-admissibility are well-known examples.
When working with a choice function, however, it is immaterial whether it is based on such a decision criterion.
The primitive objects on this approach are simply the set-valued choices themselves, and the choice function that represents all these choices serves as an uncertainty model in and by itself.

The seminal work by Seidenfeld et al.~\cite{seidenfeld2010} has shown that a strong advantage of working with choice functions is that they allow us to impose axioms on choices, aimed at characterising what it means for choices to be rational and internally consistent.
This is also what we want to do here, but we believe our angle of approach to be novel and unique: rather than think of choice intuitively, we provide it with a concrete interpretation in terms of desirability~\cite{couso2011,walley2000,cooman2010,cooman2011b} or binary preference~\cite{seidenfeld1995}.
Another important feature of our approach is that we consider a very general setting, where the options form an abstract real vector space; horse lotteries and gambles correspond to special cases.

The basic structure of our paper is as follows.
We start in Section~\ref{sec:choice:functions:and:interpretation} by introducing choice functions and our interpretation for them.
Next, in Section~\ref{sec:coherent:sets:of:option:sets}, we develop an alternative but equivalent way of describing these choice functions: sets of desirable option sets.
We use our interpretation to suggest and motivate a number of rationality, or coherence, axioms for such sets of desirable option sets, and show in Section~\ref{sec:back:to:choice} what are the corresponding coherence axioms for choice (or rejection) functions.
Section~\ref{sec:binary:choice} deals with the special case of binary choice, and its relation to the theory of sets of desirable options \cite{couso2011,walley2000,cooman2010,cooman2011b} and binary preference.
This is important because our main result in Section~\ref{sec:representation} shows that any coherent choice model can be represented in terms of sets of such binary choice models.
In the remaining Sections~\ref{sec:totality}--\ref{sec:archimedeanity}, we consider additional axioms or properties, such as totality, the mixing property, and an Archimedean property, and prove corresponding representation results.
This includes representations in terms of sets of strict total orders, sets of lexicographic probability systems, sets of coherent lower previsions and sets of linear previsions.
To facilitate the reading, proofs and intermediate results have been relegated to the Appendix.

\section{Choice functions and their interpretation}\label{sec:choice:functions:and:interpretation}

A choice function $\choicefun$ is a set-valued operator on sets of options.
In particular, for any set of options $\optset$, the corresponding value of $\choicefun$ is a subset $\choicefun\group{\optset}$ of $\optset$.
The options themselves are typically actions amongst which a subject wishes to choose. We here follow a very general approach where these options constitute an abstract real vector space $\opts$ provided with a---so-called \emph{background}---vector ordering $\optlteq$ and a strict version $\optlt$.
The elements $\opt$ of $\opts$ are called \emph{options} and $\opts$ is therefore called the \emph{option space}.
We let $\opts_{\optgt0}\coloneqq\cset{\opt\in\opts}{\opt\optgt0}$.
The purpose of a choice function is to represent our subject's choices between such options.

Our motivation for adopting this general framework where options are elements of abstract vector spaces, rather than the more familiar one that focuses on choice between, say, horse lotteries \cite{aumann1962,aumann1964,nau2006,seidenfeld1995,seidenfeld2010}, is its applicability to various contexts.

A typical set-up that is customary in decision theory, for example, is one where every option has a corresponding reward that depends on the state of a variable~$X$, about which the subject is typically uncertain.
Hence, the reward is uncertain too.
As a special case, therefore, we can consider that the variable~$X$ takes values~$x$ in some set of states~$\mathcal{X}$.
The reward that corresponds to a given option is then a function $\opt$ on $\mathcal{X}$.
If we assume that this reward can be expressed in terms of a real-valued linear utility scale, this allows us to identify every act with a real-valued map on $\mathcal{X}$.
These maps are often taken to be bounded and are then called \emph{gambles} on $X$.
In that context, we can consider the different gambles on $X$ as our options, and the vector space $\opts$ as the set of all such gambles.
Two popular vector orderings on $\opts$ then correspond to choosing
\begin{equation*}
\opts_{\optgt0}\coloneqq\cset{\opt\in\opts}{\opt\gblgteq0\text{ and }\opt\neq0}
\text{~or~}
\opts_{\optgt0}\coloneqq\cset{\opt\in\opts}{\inf\opt>0},
\end{equation*}
where $\geqslant$ represents the point-wise ordering of gambles, defined by
\begin{equation*}
\opt\geqslant\altopt\Leftrightarrow\group{\forall x\in\mathcal{X}}\opt(x)\geq\altopt(x).
\end{equation*}

A more general framework, which allows us to dispense with the linearity assumption of the utility scale, consists in considering as option space the linear space of all bounded real-valued maps on the set $\states\times\rewards$, where $\mathcal{R}$ is a (finite) set of rewards.
Zaffalon and Miranda \cite{zaffalon2017:incomplete:preferences} have shown that, in a context of binary preference relations, this leads to a theory that is essentially equivalent to the classical horse lottery approach.
It tends, however, to be more elegant, because a linear space is typically easier to work with than a convex set of horse lotteries.
Van Camp \cite{2017vancamp:phdthesis} has shown that this idea can be straightforwardly extended from binary preference relations to the more general context of choice functions.
We follow his lead in focusing on linear spaces of options here.

In both of the above-mentioned cases, the options are still bounded real-valued maps.
In fairly recent work, Van Camp et al.~\cite{2017vancamp:phdthesis,vancamp2015:indifference} have shown that a notion of indifference can be associated with choice functions quite easily, by moving from the original option space to its quotient space with respect to the linear subspace of all options that are assessed to be equivalent to the zero option.
Even when the original options are real-valued maps, the elements of the quotient space will be equivalence classes of such maps---affine subspaces of the original option space---which can no longer be straightforwardly identified with real-valued maps.
This provides even more incentives for considering options to be vectors in some abstract linear space $\opts$.

Having introduced and motivated our abstract option space $\opts$, sets of options can now be identified with subsets of $\opts$, which we call \emph{option sets}.
We restrict our attention here to \emph{finite} option sets and will use $\mathcal{Q}$ to denote the set of all such finite subsets of $\opts$, including the empty set.

\begin{definition}[Choice function]\label{def:choicefunction}
A \emph{choice function} $\choicefun$ is a map from $\mathcal{Q}$ to $\mathcal{Q}$ such that $\choicefun\group{\optset}\subseteq\optset$ for every $\optset\in\mathcal{Q}$.
\end{definition}
\noindent
Options in $\optset$ that do not belong to $\choicefun\group{\optset}$ are said to be \emph{rejected}.
This leads to an alternative but equivalent representation in terms of rejection functions: the \emph{rejection function} $\rejectfun[\choicefun]$ corresponding to a choice function $\choicefun$ is a map from $\mathcal{Q}$ to $\mathcal{Q}$, defined by $\rejectfun[\choicefun]\group{\optset}\coloneqq\optset\setminus\choicefun\group{\optset}$ for all $\optset\in\mathcal{Q}$.

Alternatively, a rejection function $\rejectfun$ can also be independently defined as a map from $\mathcal{Q}$ to $\mathcal{Q}$ such that $\rejectfun\group{\optset}\subseteq\optset$ for all $\optset\in\mathcal{Q}$.
The corresponding choice function $\choicefun[\rejectfun]$ is then clearly defined by $\choicefun[\rejectfun]\group{\optset}\coloneqq\optset\setminus\rejectfun\group{\optset}$ for all $\optset\in\mathcal{Q}$.
\noindent
Since a choice function is completely determined by its rejection function, any interpretation for rejection functions automatically implies an interpretation for choice functions.
This allows us to focus on the former.

Our interpretation for rejection functions---and therefore also for choice functions---now goes as follows.
Consider a subject whose uncertainty is represented by a rejection function $\rejectfun$, or equivalently, by a choice function $\choicefun[\rejectfun]$.
Then for a given option set $\optset\in\mathcal{Q}$, the statement that an option $\opt\in\optset$ is rejected from $\optset$---that is, that $\opt\in\rejectfun\group{\optset}$---is taken to mean that \emph{there is at least one option $\altopt$ in $\optset$ that our subject strictly prefers over $\opt$}.

If we denote the strict preference of one option $\altopt$ over another option $\opt$ by $\altopt\prefgt\opt$, this can be written succinctly as
\begin{equation}\label{eq:interpretation:rejectfuns}
\group{\forall\optset\in\mathcal{Q}}
\group{\forall\opt\in\optset}
\group[\big]{\opt\in\rejectfun\group{\optset}
\Leftrightarrow
\group{\exists\altopt\in\optset}\altopt\prefgt\opt}.
\end{equation}
In this paper, such a statement---as well as statements such as those in Equations~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity} and~\eqref{eq:interpretation:rejectfuns:after:irreflexivityandadditivity:Gert}---will be interpreted as providing information about a strict preference relation~$\prefgt$, that may or may not be known or specified.
The only requirements that we impose on~$\prefgt$ is that it should be a strict partial order that extends the background ordering~$\optgt$ and is compatible with the vector space operations on $\opts$:
\begin{enumerate}[label=$\protect{\prefgt[{\arabic*}]}$.,ref=$\protect{\prefgt[{\arabic*}]}$,start=0,leftmargin=*]
\item\label{ax:prefgt:irreflexive}
$\prefgt$ is irreflexive: for all $\opt\in\opts$, $\opt\not\prefgt\opt$;
\item\label{ax:prefgt:transitive} $\prefgt$ is transitive: for all $\opt,\altopt,\altopttoo\in\opts$, $\opt\prefgt\altopt$ and $\altopt\prefgt\altopttoo$ imply that also $\opt\prefgt\altopttoo$;
\item\label{ax:prefgt:background} for all $\opt,\altopt\in\opts$, $\opt\optgt\altopt$ implies that $\opt\prefgt\altopt$;
\item\label{ax:prefgt:addition}
for all $\opt,\altopt,\altopttoo\in\opts$, $\opt\prefgt\altopt$ implies that---so is equivalent with---$\opt+\altopttoo\prefgt\altopt+\altopttoo$;
\item\label{ax:prefgt:multiplication} for all $\opt,\altopt\in\opts$ and all $\lambda>0$, $\opt\prefgt\altopt$ implies that---so is equivalent with---$\lambda\opt\prefgt\lambda\altopt$.
\end{enumerate}
We then call such a preference ordering~$\prefgt$ \emph{coherent}.
It follows from Axioms~\ref{ax:prefgt:irreflexive} and~\ref{ax:prefgt:addition} that we can rewrite Equation~\eqref{eq:interpretation:rejectfuns} as
\begin{equation}\label{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity}
\group{\forall\optset\in\mathcal{Q}}
\group{\forall\opt\in\optset}
\group[\Big]{\opt\in\rejectfun\group{\optset}
\Leftrightarrow
\group{\exists\altopt\in\optset}\altopt-\opt\prefgt0
\Leftrightarrow
\group{\exists\altopt\in\optset\setminus\set{\opt}}\altopt-\opt\prefgt0},
\end{equation}
where we use Axiom~\ref{ax:prefgt:addition} for the first equivalence, and Axiom~\ref{ax:prefgt:irreflexive} for the second.
Both equivalences can be conveniently turned into a single one if we no longer require that $\opt$ should belong to $\optset$ and consider statements of the form $\opt\in\rejectfun\group{\optset\cup\set{u}}$.
Equation~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity} then turns into
\begin{equation}\label{eq:interpretation:rejectfuns:after:irreflexivityandadditivity:Gert}
\group{\forall\opt\in\opts}\group{\forall\optset\in\mathcal{Q}}
\Big(\opt\in\rejectfun\group{\optset\cup\set{\opt}}
\Leftrightarrow
\group{\exists\altopt\in\optset}\altopt-\opt\prefgt0\Big),
\end{equation}
So, according to our interpretation, the statement that $\opt$ is rejected from $\optset\cup\set{\opt}$ is taken to mean that the option set
\begin{equation}\label{eq:specialminus}
\optset-\opt\coloneqq\cset{\altopt-\opt}{\altopt\in\optset}
\end{equation}
contains at least one option that, according to~ $\prefgt$, is strictly preferred to the zero option~$0$.

\section{Coherent sets of desirable option sets}\label{sec:coherent:sets:of:option:sets}

A crucial observation at this point is that our interpretation for rejection functions does not require our subject to specify the strict preference $\prefgt$.
Instead, all that is needed is for her to specify option sets $\optset\in\mathcal{Q}$ that---to her---contain at least one option that is strictly preferred to the zero option $0$.
Options that are strictly preferred to zero---so options $\opt$ for which $\opt\prefgt0$---are also called \emph{desirable}, which is why we will call such option sets \emph{desirable option sets} and collect them in a \emph{set of desirable option sets} $\rejectset\subseteq\mathcal{Q}$.
Our interpretation therefore allows a modeller to specify her beliefs by specifying a \emph{set of desirable option sets} $\rejectset\subseteq\mathcal{Q}$.

As can be seen from Equations~\eqref{eq:interpretation:rejectfuns:after:irreflexivityandadditivity:Gert} and~\eqref{eq:specialminus}, such a set of desirable option sets $\rejectset$ completely determines a rejection function $\rejectfun$ and its corresponding choice function $\choicefun[\rejectfun]$:
\begin{equation}\label{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert}
\group{\forall\opt\in\opts}
\group{\forall\optset\in\mathcal{Q}}
\group[\big]{\opt\in\rejectfun\group{\optset\cup\set{\opt}}
\Leftrightarrow\optset-\opt\in\rejectset}.
\end{equation}
\emph{Our interpretation, together with the basic Axioms~\ref{ax:prefgt:irreflexive} and~\ref{ax:prefgt:addition}, therefore allows the study of rejection and choice functions to be reduced to the study of sets of desirable option sets.}

We let $\mathbf{K}$ denote the set of all sets of desirable option sets $\rejectset\subseteq\mathcal{Q}$, and consider any such~$\rejectset\in\mathbf{K}$.
The first question to address is when to call $\rejectset$ \emph{coherent}: which properties should we impose on a set of desirable option sets in order for it to reflect a rational subject's beliefs? We propose the following axiomatisation, using $(\lambda,\mu)>0$ as a shorthand notation for `$\lambda\geq0$, $\mu\geq0$ and $\lambda+\mu>0$'.

\begin{definition}[Coherence for sets of desirable option sets]\label{def:coherence:rejectset}
A set of desirable option sets $\rejectset\subseteq\mathcal{Q}$ is called \emph{coherent} if it satisfies the following axioms:
\begin{enumerate}[label=$\mathrm{K}_{\arabic*}$.,ref=$\mathrm{K}_{\arabic*}$,leftmargin=*,start=0]
\item\label{ax:rejects:removezero} if $\optset\in\rejectset$ then also $\optset\setminus\set{0}\in\rejectset$, for all $\optset\in\mathcal{Q}$;
\item\label{ax:rejects:nozero} $\set{0}\notin\rejectset$;
\item\label{ax:rejects:pos} $\set{\opt}\in\rejectset$, for all $\opt\in\opts_{\optgt0}$;
\item\label{ax:rejects:cone} if $\optset[1],\optset[2]\in\rejectset$ and if, for all $\opt\in\optset[1]$ and $\altopt\in\optset[2]$, $(\lambda_{\opt,\altopt},\mu_{\opt,\altopt})>0$, then also\footnote{The following simple example might help the reader understand what this axiom allows for. Consider any two $a,b\in\opts$, let $A_1=A_2=A\coloneqq\{a,b\}$ and choose $(\lambda_{a,a},\mu_{a,a})=(1,0)$, $(\lambda_{a,b},\mu_{a,b})=(1,1)$, $(\lambda_{b,a},\mu_{b,a})=(1,1)$ and $(\lambda_{b,b},\mu_{b,b})=(1,1)$. Then if $A\in\rejectset$, it follows from Axiom~\ref{ax:rejects:cone} that also $\{a,a+b,2b\}\in\rejectset$.}
\begin{equation*}
\cset{\lambda_{\opt,\altopt}\opt+\mu_{\opt,\altopt}\altopt}{\opt\in\optset[1],\altopt\in\optset[2]}
\in\rejectset;
\end{equation*}
\item\label{ax:rejects:mono} if $\optset[1]\in\rejectset$ and $\optset[1]\subseteq\optset[2]$, then also $\optset[2]\in\rejectset$, for all $\optset[1],\optset[2]\in\mathcal{Q}$.
\end{enumerate}
We denote the set of all coherent sets of desirable option sets by $\overline{\rejectsets}$.
\end{definition}
\noindent
This axiomatisation is entirely based on our interpretation and the following three axioms for desirability:
\begin{enumerate}[label=$\mathrm{d}_{\arabic*}$.,ref=$\mathrm{d}_{\arabic*}$,start=1,leftmargin=*]
\item\label{ax:desirability:nozero}
$0$ is not desirable;
\item\label{ax:desirability:pos} all $\opt\in\opts_{\optgt0}$ are desirable;
\item\label{ax:desirability:cone} if $\opt,\altopt$ are desirable and $(\lambda,\mu)>0$, then $\lambda\opt+\mu\altopt$ is desirable.
\end{enumerate}
Each of these three axioms follows trivially from our assumptions on the preference relation $\prefgt$: \ref{ax:desirability:nozero} follows from~\ref{ax:prefgt:irreflexive}, \ref{ax:desirability:pos} follows from~\ref{ax:prefgt:background} and~\ref{ax:desirability:cone} follows from~\ref{ax:prefgt:transitive} and~\ref{ax:prefgt:multiplication}.\footnote{Conversely, under Axiom~\ref{ax:prefgt:addition} for the preference relation~$\prefgt$, the three Axioms~\ref{ax:desirability:nozero}--\ref{ax:desirability:cone} imply the remaining Axioms~\ref{ax:prefgt:irreflexive}--\ref{ax:prefgt:background} and~\ref{ax:prefgt:multiplication}.}

That the coherence Axioms~\ref{ax:rejects:removezero}--\ref{ax:rejects:mono} are implied by our rationality requirements~\ref{ax:desirability:nozero}--\ref{ax:desirability:cone} for the concept of desirability, can now be seen as follows.
Since a desirable option set is by definition a set of options that contains at least one desirable option, Axiom~\ref{ax:rejects:mono} is immediate.
Axioms~\ref{ax:rejects:removezero} and~\ref{ax:rejects:nozero} follow naturally from \ref{ax:desirability:nozero}, and Axiom~\ref{ax:rejects:pos} is an immediate consequence of \ref{ax:desirability:pos}.
The argument for Axiom~\ref{ax:rejects:cone} is more subtle.
Since $\optset[1]$ and $\optset[2]$ are two desirable option sets, there must be at least one desirable option $\opt\in\optset[1]$ and one desirable option $\altopt\in\optset[2]$.
Since for these two options, the positive linear combination $\lambda_{\opt,\altopt}\opt+\mu_{\opt,\altopt}\altopt$ is again desirable by \ref{ax:desirability:cone}, at least one of the elements of the option set $\cset{\lambda_{\opt,\altopt}\opt+\mu_{\opt,\altopt}\altopt}{\opt\in\optset[1],\altopt\in\optset[2]}$ must be a desirable option.
Hence, it must be a desirable option set.

\section{Coherent rejection functions}\label{sec:back:to:choice}

Now that we have formulated our basic rationality requirements~\ref{ax:rejects:removezero}--\ref{ax:rejects:mono} for sets of desirable option sets $\rejectset$, we are in a position to use their correspondence~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert} with rejection  functions~$\rejectfun$ to derive the equivalent rationality requirements for the latter.

Equation~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert} already allows us to derive a first and very basic axiom for rejection functions---and a very similar one for choice functions, left implicit here---without imposing \emph{any} requirements on sets of desirable option sets~$\rejectset$:
\begin{enumerate}[label=$\mathrm{R}_{\arabic*}$.,ref=$\mathrm{R}_{\arabic*}$,start=0,leftmargin=*]
\item\label{ax:rejectfun:addition} for all $\optset\in\mathcal{Q}$ and $\opt\in\optset$, $\opt\in\rejectfun\group{\optset}$ if and only if $0\in\rejectfun\group{\optset-\opt}$.
\end{enumerate}
Alternatively, we can also consider a slightly different---but clearly equivalent---version that perhaps displays better the invariance of rejection functions under vector addition:
\begin{enumerate}[label=$\mathrm{R}_{\arabic*}'$.,ref=$\mathrm{R}_{\arabic*}'$,start=0,leftmargin=*]
\item\label{ax:rejectfun:addition:too} for all $\optset\in\mathcal{Q}$, $\opt\in\optset$ and $\altopt\in\opts$, $\opt\in\rejectfun\group{\optset}$ if and only if $\opt+\altopt\in\rejectfun\group{\optset+\altopt}$.
\end{enumerate}

When we do impose requirements on sets of desirable option sets~$\rejectset$, Equation~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert} allows us to turn them into requirements for rejection (and hence also choice) functions.
In particular, we will see in Proposition~\ref{prop:axioms:rejection:sets:and:functions:coherence} below that our Axioms~\ref{ax:rejects:removezero}--\ref{ax:rejects:mono} imply that
\begin{enumerate}[label=$\mathrm{R}_{\arabic*}$.,ref=$\mathrm{R}_{\arabic*}$,leftmargin=*]
\item\label{ax:rejectfun:not:everything:rejected} $\rejectfun\group{\emptyset}=\emptyset$, and $\rejectfun\group{\optset}\neq\optset$ for all $\optset\in\mathcal{Q}\setminus\set{\emptyset}$;
\item\label{ax:rejectfun:pos} $0\in\rejectfun\group{\set{0,\opt}}$, for all $\opt\in\opts_{\optgt0}$;
\item\label{ax:rejectfun:cone} if $\optset[1],\optset[2]\in\mathcal{Q}$ such that $0\in\rejectfun\group{\optset[1]\cup\set{0}}$ and $0\in\rejectfun\group{\optset[2]\cup\set{0}}$ and if $(\lambda_{\opt,\altopt},\mu_{\opt,\altopt})>0$ for all $\opt\in\optset[1]$ and $\altopt\in\optset[2]$, then also
\begin{equation*}
0\in\rejectfun\group{\cset{\lambda_{\opt,\altopt}\opt+\mu_{\opt,\altopt}\altopt}{\opt\in\optset[1],\altopt\in\optset[2]}\cup\set{0}};
\end{equation*}
\item\label{ax:rejectfun:senalpha}
if $\optset[1]\subseteq\optset[2]$ then also $\rejectfun\group{\optset[1]}\subseteq\rejectfun\group{\optset[2]}$, for all $\optset[1],\optset[2]\in\mathcal{Q}$.
\end{enumerate}
Axiom~\ref{ax:rejectfun:senalpha} is Sen's condition~$\alpha$ \cite{sen1971,sen1977}.
Arthur Van Camp (private communication) has proved in a direct manner that Aizermann's condition \cite{aizerman1985} can be derived from our Axioms~\ref{ax:rejects:removezero}--\ref{ax:rejects:mono} as well.
Indirectly, this can also be inferred from our representation results further on; see Theorem~\ref{theo:coherentrepresentation:twosided} in Section~\ref{sec:representation}, and the discussion following it.

We will call \emph{coherent} any rejection function that satisfies the five properties~\ref{ax:rejectfun:addition}--\ref{ax:rejectfun:senalpha} above.

\begin{definition}[Coherence for rejection and choice functions]\label{def:coherence:rejectfun}
A rejection function $\rejectfun$ is called \emph{coherent} if it satisfies the Axioms~\ref{ax:rejectfun:addition}--\ref{ax:rejectfun:senalpha}.
A choice function $\choicefun$ is called \emph{coherent} if the associated rejection function $\rejectfun[\choicefun]$ is.
\end{definition}

Our next result establishes that these notions of coherence are perfectly compatible with the coherence for sets of desirable option sets that we introduced in Section~\ref{sec:coherent:sets:of:option:sets}.

\begin{proposition}\label{prop:axioms:rejection:sets:and:functions:coherence}
Consider any set of desirable option sets $\rejectset\in\mathbf{K}$ and any rejection function $\rejectfun$ that are connected by Equation~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert}.
Then $\rejectset$ is coherent if and only if $\rejectfun$ is.
\end{proposition}

We will from now on work directly with (coherent) sets of desirable option sets and will use the collective term \emph{(coherent) choice models} for (coherent) choice functions, rejection functions, and sets of desirable option sets.
Of course, our primary motivation for studying coherent sets of desirable option sets is their connection with the other two choice models.
This being said, it should however also be clear that our results do not depend on this connection.
The theory of sets of desirable option sets that we are about to develop can therefore be used independently as well.

\section{The special case of binary choice}\label{sec:binary:choice}

According to our interpretation, the statement that $\optset$ belongs to a set of desirable option sets $\rejectset$ is taken to mean that $\optset$ contains at least one desirable option.
This implies that singletons play a special role: for any $\opt\in\opts$, stating that $\set{\opt}\in\rejectset$ is equivalent to stating that $\opt$ is desirable. For any set of desirable option sets $\rejectset$, these singleton assessments are captured completely by the set of options
\begin{equation}\label{eq:desir:from:choice}
\desirset[\rejectset]\coloneqq\cset{\opt\in\opts}{\set{\opt}\in\rejectset}
\end{equation}
that, according to $\rejectset$, are definitely desirable---preferred to $0$.
A set of desirable option sets $\rejectset\in\mathbf{K}$ that is completely determined by such singleton assessments is called \emph{binary}.

\begin{definition}[Binary set of desirable option sets]\label{def:binary}
We call a set of desirable option sets $\rejectset$ \emph{binary} if
\vspace{-5pt}
\begin{equation}\label{eq:binary}
\optset\in\rejectset\Leftrightarrow(\exists\opt\in\optset)\set{\opt}\in\rejectset,
\text{ for all $\optset\in\mathcal{Q}$}.
\end{equation}
\end{definition}

In order to explain how any binary set of desirable option sets $\rejectset$ is indeed completely determined by $\desirset[\rejectset]$, we need a way to associate a rejection function with sets of options such as $\desirset[\rejectset]$. To that end, we consider the notion of a \emph{set of desirable options}: a subset $\desirset$ of $\opts$ whose interpretation will be that it consists of the options $\opt\in\opts$ that our subject considers desirable. We denote the set of all such sets of desirable options $\desirset\subseteq\opts$ by $\mathbf{D}$.

With any $\desirset\in\mathbf{D}$, our interpretation for rejection functions in Section~\ref{sec:choice:functions:and:interpretation} inspires us to associate a set of desirable option sets $\rejectset[\desirset]$, defined by
\begin{equation}\label{eq:desirset:to:rejectset}
\rejectset[\desirset]
\coloneqq\cset{\optset\in\mathcal{Q}}{\optset\cap\desirset\neq\emptyset}.
\end{equation}
It turns out that a set of desirable options sets $\rejectset$ is binary if and only if it has the form $\rejectset[\desirset]$, and the unique \emph{representing} $\desirset$ is then given by $\desirset[\rejectset]$.

\begin{proposition}\label{prop:binary:characterisation}
A set of desirable options sets $\rejectset\in\mathbf{K}$ is binary if and only if there is some~$\desirset\in\mathbf{D}$ such that $\rejectset=\rejectset[\desirset]$.
This $\desirset$ is then necessarily unique, and equal to $\desirset[\rejectset]$.
\end{proposition}

Just like we did for sets of desirable option sets in Section~\ref{sec:coherent:sets:of:option:sets}, we can use the basic rationality principles~\ref{ax:desirability:nozero}--\ref{ax:desirability:cone} for the notion of desirability---or binary preference---to infer basic rationality criteria for sets of desirable options.
When they do, we call them coherent.

\begin{definition}[Coherence for sets of desirable options]\label{def:cohdesir}
A set of desirable options $\desirset\in\mathbf{D}$ is called \emph{coherent} if it satisfies the following axioms:\footnote{The Axioms~\ref{ax:desirs:nozero}--\ref{ax:desirs:cone} for sets of desirable options should not be confused with the rationality criteria~\ref{ax:desirability:nozero}--\ref{ax:desirability:cone} for our primitive notion of desirability---or binary preference. Like the Axioms~\ref{ax:rejects:removezero}--\ref{ax:rejects:mono}, they are only derived from these primitive assumptions on the basis of their interpretation.}
\begin{enumerate}[label=$\mathrm{D}_{\arabic*}$.,ref=$\mathrm{D}_{\arabic*}$,leftmargin=*]
\item\label{ax:desirs:nozero} $0\notin\desirset$;
\item\label{ax:desirs:pos} $\opts_{\optgt0}\subseteq\desirset$;
\item\label{ax:desirs:cone} if $\opt,\altopt\in\desirset$ and $(\lambda,\mu)>0$, then $\lambda\opt+\mu\altopt\in\desirset$.
\end{enumerate}
We denote the set of all coherent sets of desirable options by $\overline{\desirsets}$.
\end{definition}
\noindent
So a coherent set of desirable options is a convex cone [Axiom~\ref{ax:desirs:cone}] in $\opts$ that does not contain $0$ [Axiom~\ref{ax:desirs:nozero}] and includes~$\opts_{\optgt0}$ [Axiom~\ref{ax:desirs:pos}].
Sets of desirable options are an abstract version of the sets of desirable gambles that have an important part in the literature on imprecise probability models \cite{couso2011,cooman2010,quaeghebeur2012:itip,walley2000}.
This abstraction was first introduced and studied in great detail in~\cite{cooman2011b,quaeghebeur2015:statement}.

Our next result shows that the coherence of a binary set of desirable option sets is completely determined by the coherence of its corresponding set of desirable options.

\begin{proposition}\label{prop:coherence:for:binary}
Consider any binary set of desirable option sets $\rejectset\in\mathbf{K}$ and let $\desirset[\rejectset]\in\mathbf{D}$ be its corresponding set of desirable options.
Then $\rejectset$ is coherent if and only if $\desirset[\rejectset]$~is.
Conversely, consider any set of desirable options $\desirset\in\mathbf{D}$ and let $\rejectset[\desirset]$ be its corresponding binary set of desirable option sets, then $\rejectset[\desirset]$ is coherent if and only if $\desirset$ is.
\end{proposition}
\noindent
So the binary coherent sets of desirable option sets are given by $\cset{\rejectset[\desirset]}{\desirset\in\overline{\desirsets}}$, allowing us to call any coherent set of desirable option sets in $\overline{\rejectsets}\setminus\cset{\rejectset[\desirset]}{\desirset\in\overline{\desirsets}}$ \emph{non-binary}.

What makes coherent sets of desirable options $\desirset\in\overline{\desirsets}$---and hence also coherent binary sets of desirable option sets---particularly interesting is that they induce a binary preference order $\prefgt[\desirset]$---a strict vector ordering---on $\opts$, defined by
\begin{equation}\label{eq:desirset:to:prefgt}
\opt\prefgt[\desirset]\altopt\Leftrightarrow\opt-\altopt\in\desirset
\text{ for all $\opt,\altopt\in\opts$}.
\end{equation}
The preference order $\prefgt[\desirset]$ is coherent---satisfies Axioms~\ref{ax:prefgt:irreflexive}--\ref{ax:prefgt:multiplication}--- and furthermore fully characterises $\desirset$: one can easily see that $\opt\in\desirset$ if and only if $\opt\prefgt[\desirset]0$.
Hence, coherent sets of desirable options and coherent binary sets of desirable option sets are completely determined by a single binary strict preference order between options. This is of course the reason why we reserve the moniker \emph{binary} for choice models that are essentially based on \emph{singleton} assessments.

\section{Representation in terms of sets of desirable options}\label{sec:representation}

It should be clear---and it should be stressed---at this point that making a direct desirability assessment for an option $\opt$ typically requires more of a subject than making a typical desirability assessment for an option set $\optset$: the former requires that our subject should state that $\opt$ is desirable, while the latter only requires the subject to state that some option in $\optset$ is desirable, but not to specify which.
It is this difference---this greater latitude in making assessments---that guarantees that our account of choice is much richer than one that is purely based on binary preference. In the framework of sets of desirable option sets, it is for instance possible to express the belief that \emph{at least} one of two options $\opt$ or $\altopt$ is desirable, while remaining undecided about which of them actually is; in order to express this belief, it suffices to state that $\set{\opt,\altopt}\in\rejectset$.
This is not possible in the framework of sets of desirable options. Sets of desirable option sets therefore constitute a much more general uncertainty framework than sets of desirable options.

So while it is nice that there are sets of desirable option sets $\rejectset[\desirset]$ that are completely determined by a set of desirable options $\desirset$, such binary choice models are typically \emph{not} what we are interested in here: using $\rejectset[\desirset]$ is equivalent to using $\desirset$ here, so there is no benefit in using the more convoluted model $\rejectset[\desirset]$ to represent choice.
No, it is the \emph{non-binary} coherent choice models that we have in our sights.
If we replace such a non-binary coherent set of desirable option sets $\rejectset$ by its corresponding set of desirable options $\desirset[\rejectset]$, we lose information, because then necessarily $\rejectset[{\desirset[\rejectset]}]\subset\rejectset$.
Choice models are therefore more expressive than sets of desirable options.
But it turns out that our coherence axioms lead to a representation result that allows us to still use sets of desirable options, or rather, sets of them, to completely characterise \emph{any} coherent choice model.

\begin{theorem}\label{theo:coherentrepresentation:twosided}
A set of desirable option sets $\rejectset\in\mathbf{K}$ is coherent if and only if there is a non-empty set $\mathcal{D}\subseteq\overline{\desirsets}$ of coherent sets of desirable options such that $\rejectset=\bigcap\cset{\rejectset[\desirset]}{\desirset\in\mathcal{D}}$.
The largest such set $\mathcal{D}$ is then\/ $\overline{\desirsets}\group{\rejectset}\coloneqq\cset{\desirset\in\overline{\desirsets}}{\rejectset\subseteq\rejectset[\desirset]}$.
\end{theorem}
\noindent
Due to the one-to-one correspondence between coherent sets of desirable options~$\desirset$ and coherent preference orders~$\prefgt[\desirset]$, this representation result tells us that working with a coherent set of desirable option sets $\rejectset$ is equivalent to working with the set of those coherent preference orders~$\prefgt[\desirset]$ for which $\rejectset\subseteq\rejectset[\desirset]$.
For the rejection function $\rejectfun$ that corresponds to $\rejectset$ through Equation~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert}, $\opt\in\rejectfun(\optset)$ means that $\opt$ is dominated in $\optset$ for all these representing coherent preference orders $\prefgt[\desirset]$.
Similarly, $\opt\in\choicefun[\rejectfun](\optset)$ means that $\opt$ is undominated according to at least one of these representing coherent preference orders $\prefgt[\desirset]$.
This effectively tells us that our coherence axioms~\ref{ax:rejects:removezero}--\ref{ax:rejects:mono} for choice models characterise a generalised type of choice under Levi's notion of E-admissibility \cite{levi1980a,troffaes2007,vancamp2015:indifference}, but with representing preference orders $\prefgt[\desirset]$ that need not be total orders based on comparing expectations.

Interestingly, any potential property of sets of desirable option sets that is preserved under taking arbitrary intersections, and that the binary choice models satisfy, is inherited from the binary models through the representation result of Theorem~\ref{theo:coherentrepresentation:twosided}.
It is easy to see that this applies in particular to Aizermann's condition \cite{aizerman1985}.

\section{Imposing totality}\label{sec:totality}

We have just shown that every coherent choice model~$\rejectset$ can be represented by a collection of coherent sets of desirable options~$\desirset$.
This leads us to wonder whether it is possible to achieve representation using only particular types of coherent~$\desirset$, and, if yes, for which types of coherent sets of desirable option sets~$\rejectset$---and hence for which types of rejection functions~$\rejectfun$ and choice functions~$\choicefun$---this is possible.
In this section, we clear the air by starting with a rather simple case, where we restrict attention to \emph{total} sets of desirable options~$\desirset$, corresponding to total preference orders $\prefgt[\desirset]$.

\begin{definition}[Totality for sets of desirable options]\label{def:totaldesirs}
We call a set of desirable options $\desirset\in\mathbf{D}$ \emph{total} if it is coherent and
\begin{enumerate}[label=$\mathrm{D}_{\mathrm{T}}$.,ref=$\mathrm{D}_{\mathrm{T}}$,leftmargin=*]
\item\label{ax:desirs:totality} for all $\opt\in\opts\setminus\set{0}$, either $\opt\in\desirset$ or $-\opt\in\desirset$.
\end{enumerate}
The set of all total sets of desirable options is denoted by $\overline{\desirsets}_{\mathrm{T}}$.
\end{definition}
\noindent
That the binary preference order $\prefgt[\desirset]$ corresponding to a total set of desirable options $\desirset$ is indeed a total order can be seen as follows.
For all $\opt,\altopt\in\opts$ such that $\opt\neq\altopt$, the property~\ref{ax:desirs:totality} implies that either $\opt-\altopt\in\desirset$ or $\altopt-\opt\in\desirset$.
Hence, for all $\opt,\altopt\in\opts$, we have that either $\opt=\altopt$, $\opt\prefgt[\desirset]\altopt$ or $\altopt\prefgt[\desirset]\opt$, which indeed makes $\prefgt[\desirset]$ a total order.

It was shown in~\cite{couso2011,cooman2010} that what we call \emph{total} sets of desirable options here, are precisely the \emph{maximal} or undominated coherent ones, i.e.~those coherent $\desirset\in\overline{\desirsets}$ that are not included in any other coherent set of desirable option sets: $\group{\forall\desirset'\in\overline{\desirsets}}\group{\desirset\subseteq\desirset'\Rightarrow\desirset=\desirset'}$.
The question of which types of binary sets of desirable option sets $\rejectset[\desirset]$ the total $\desirset$ correspond to, is answered by the following definition and proposition.

\begin{definition}[Totality for sets of desirable option sets]\label{def:totalrejects}
We call a set of desirable option sets $\rejectset\in\mathbf{K}$ \emph{total} if it is coherent and
\begin{enumerate}[label=$\mathrm{K}_{\mathrm{T}}$.,ref=$\mathrm{K}_{\mathrm{T}}$,leftmargin=*]
\item\label{ax:rejects:totality}
$\set{\opt,-\opt}\in\rejectset$ for all $\opt\in\opts\setminus\set{0}$.
\end{enumerate}
The set of all total sets of desirable options is denoted by $\overline{\rejectsets}_{\mathrm{T}}$.
\end{definition}

\begin{proposition}\label{prop:totalbinaryiff}
For any set of desirable options $\desirset\in\mathbf{D}$, $\desirset$ is total if and only if~$\rejectset[\desirset]$ is, so $\rejectset[\desirset]\in\overline{\rejectsets}_{\mathrm{T}}\Leftrightarrow\desirset\in\overline{\desirsets}_{\mathrm{T}}$.
\end{proposition}
\noindent
So a binary $\rejectset$ is total if and only if its corresponding $\desirset[\rejectset]$ is.
For general total sets of desirable option sets $\rejectset\in\overline{\rejectsets}_{\mathrm{T}}$, which are not necessarily binary, we nevertheless still have representation in terms of total binary ones.

\begin{theorem}\label{theo:totalrepresentation:twosided}
A set of desirable option sets $\rejectset\in\mathbf{K}$ is total if and only if there is a non-empty set $\mathcal{D}\subseteq\overline{\desirsets}_{\mathrm{T}}$ of total sets of desirable options such that $\rejectset=\bigcap\cset{\rejectset[\desirset]}{\desirset\in\mathcal{D}}$.
The largest such set $\mathcal{D}$ is then\/ $\overline{\desirsets}_{\mathrm{T}}\group{\rejectset}\coloneqq\cset{\desirset\in\overline{\desirsets}_{\mathrm{T}}}{\rejectset\subseteq\rejectset[\desirset]}$.
\end{theorem}
\noindent
This representation result shows that our total choice models correspond a generalised type of choice under Levi's notion of E-admissibility \cite{levi1980a,troffaes2007}, but with representing preference orders $\prefgt[\desirset]$ that are now maximal, or undominated.
They correspond what Van Camp et al.~\cite[Section~4]{vancamp2015:indifference} have called \emph{M-admissible} choice models.
Our discussion above provides an axiomatic characterisation for such choice models.

We conclude our study of totality by characterising what it means for a rejection function to be total.

\begin{proposition}\label{prop:totalityforrejectionfunctions}
Consider any set of desirable option sets~$\rejectset\in\mathbf{K}$ and any rejection function~$\rejectfun$ that are connected by Equation~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert}.
Then $\rejectset$ is total if and only if $\rejectfun$ is coherent and satisfies
\begin{enumerate}[label=$\mathrm{R}_{\mathrm{T}}$.,ref=$\mathrm{R}_{\mathrm{T}}$,leftmargin=*]
\item\label{ax:rejectfun:total} $0\in\rejectfun\group{\set{0,\opt,-\opt}}$, for all $\opt\in\opts\setminus\set{0}$.
\end{enumerate}
\end{proposition}

\section{Imposing the mixing property}\label{sec:mixture}

Totality is, of course, a very strong requirement, and it leads to a very special and restrictive type of representation.
We therefore now turn to weaker requirements, and their consequences for representation.
One such additional property, which sometimes pops up in the literature about choice and rejection functions, is the following \emph{mixing property} \cite{seidenfeld2010,2017vancamp:phdthesis}, which asserts that an option that is rejected continues to be rejected if one removes mixed options---convex combinations of other options in the option set:\footnote{Van Camp \cite{2017vancamp:phdthesis} refers to this property as `convexity', but we prefer to stick to the original name suggested by Seidenfeld et al.~\cite{seidenfeld2010} for the sake of avoiding confusion. We nevertheless want to point out that in a context that focuses on rejection rather than choice, the term `\emph{unmixing}' would be preferable, because the rejection is preserved under \emph{removing} mixed options---whereas the choice is preserved under \emph{adding} mixed options.}
\begin{enumerate}[label=$\mathrm{R}_{\mathrm{M}}$.,ref=$\mathrm{R}_{\mathrm{M}}$,leftmargin=*,start=5]
\item\label{ax:rejectfun:removepositivecombinations}
if $\optset\subseteq\altoptset\subseteq\chull\group{\optset}$ then also $\rejectfun\group{\altoptset}\cap\optset\subseteq\rejectfun\group{\optset}$, for all $\optset,\altoptset\in\mathcal{Q}$,
\end{enumerate}
where $\chull\group{\cdot}$ is the \emph{convex hull operator}, defined by
\begin{equation}\label{eq:convexhulloperator}
\chull(V)
\coloneqq\cset[\bigg]{\sum_{k=1}^n\lambda_k\opt[k]}
{n\in\mathbb{N},\lambda_k\in\reals_{>0},\sum_{k=1}^n\lambda_k=1,\opt[k]\in V}
\text{ for all $V\subseteq\opts$}.
\end{equation}
$\mathbb{N}$ is the set of natural numbers, or in other words all positive integers, excluding~$0$, and $\reals_{>0}$ is the set of all (strictly) positive reals.
A rejection function that satisfies this mixing property is called \emph{mixing}.

The following result characterises the mixing property in terms of the corresponding set of desirable option sets.
We provide two equivalent conditions: one in terms of the convex hull operator, and one in terms of the $\posi\group{\cdot}$ operator, which, for any subset~$V$ of~$\opts$, returns the set of all positive linear combinations of its elements:
\begin{equation}\label{eq:posioperator}
\posi(V)\coloneqq\cset[\bigg]{\sum_{k=1}^n\lambda_k\opt[k]}{n\in\mathbb{N},\lambda_k\in\reals_{>0},\opt[k]\in V}.
\end{equation}

\begin{proposition}\label{prop:axioms:rejection:sets:and:functions:mixing}
Consider any set of desirable option sets~$\rejectset\in\mathbf{K}$ and any rejection function~$\rejectfun$ that are connected by Equation~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert}.
Then $\rejectfun$ is coherent and mixing if and only if~$\rejectset$ is coherent and satisfies any---and hence both---of the following conditions:
\begin{enumerate}[label=$\mathrm{K}_{\mathrm{M}}$.,ref=$\mathrm{K}_{\mathrm{M}}$,leftmargin=*]
\item\label{ax:rejects:removepositivecombinations} if $\altoptset\in\rejectset$ and $\optset\subseteq\altoptset\subseteq\posi\group{\optset}$, then also $\optset\in\rejectset$, for all $\optset,\altoptset\in\mathcal{Q}$;
\end{enumerate}
\begin{enumerate}[label=$\mathrm{K}'_{\mathrm{M}}$.,ref=$\mathrm{K}'_{\mathrm{M}}$,leftmargin=*]
\item\label{ax:rejects:removeconvexcombinations} if $\altoptset\in\rejectset$ and $\optset\subseteq\altoptset\subseteq\chull\group{\optset}$, then also $\optset\in\rejectset$, for all $\optset,\altoptset\in\mathcal{Q}$.
\end{enumerate}
\end{proposition}
\noindent
In the context of sets of desirable options in linear spaces, we prefer to use the $\posi\group{\cdot}$ operator because it fits more naturally with Axiom~\ref{ax:desirability:cone}.
We therefore adopt Axiom~\ref{ax:rejects:removepositivecombinations} as our definition for mixingness.

\begin{definition}[Mixing property for sets of desirable option sets]\label{def:mixingrejects}
We call a set of desirable option sets $\rejectset\in\mathbf{K}$ \emph{mixing} if it is coherent and satisfies \ref{ax:rejects:removepositivecombinations}.
The set of all mixing sets of desirable options is denoted by $\overline{\rejectsets}_{\mathrm{M}}$.
\end{definition}

We now proceed to show that these mixing sets of desirable option sets allow for a representation in terms of sets of desirable options that are themselves mixing, in the following sense.

\begin{definition}[Mixing property for sets of desirable options]\label{def:mixingdesirs}
We call a set of desirable options $\desirset\in\mathbf{D}$ \emph{mixing} if it is coherent and
\begin{enumerate}[label=$\mathrm{D}_{\mathrm{M}}$.,ref=$\mathrm{D}_{\mathrm{M}}$,leftmargin=*]
\item for all $\optset\in\mathcal{Q}$, if $\posi(\optset)\cap\desirset\neq\emptyset$, then also $\optset\cap\desirset\neq\emptyset$.\label{ax:desirs:mixing}
\end{enumerate}
We denote the set of all mixing sets of desirable options by $\overline{\desirsets}_{\mathrm{M}}$.
\end{definition}

The \emph{binary} elements of $\overline{\rejectsets}_{\mathrm{M}}$ are precisely the ones based on such a mixing set of desirable options; they can be represented by a single element of $\overline{\desirsets}_{\mathrm{M}}$.

\begin{proposition}\label{prop:mixingbinaryiffD}
For any set of desirable options $\desirset\in\mathbf{D}$, $\rejectset[\desirset]$ is mixing if and only if~$\desirset$ is, so $\rejectset[\desirset]\in\overline{\rejectsets}_{\mathrm{M}}\Leftrightarrow\desirset\in\overline{\desirsets}_{\mathrm{M}}$.
\end{proposition}
\noindent
For general mixing sets of desirable option sets that are not necessarily binary, we nevertheless still obtain a representation theorem analogous to Theorems~\ref{theo:coherentrepresentation:twosided} and~\ref{theo:totalrepresentation:twosided}, where the representing sets of desirable options are now mixing.

\begin{theorem}\label{theo:mixingrepresentation:twosided}
A set of desirable option sets $\rejectset\in\mathbf{K}$ is mixing if and only if there is a non-empty set $\mathcal{D}\subseteq\overline{\desirsets}_{\mathrm{M}}$ of mixing sets of desirable options such that $\rejectset=\bigcap\cset{\rejectset[\desirset]}{\desirset\in\mathcal{D}}$.
The largest such set $\mathcal{D}$ is then\/ $\overline{\desirsets}_{\mathrm{M}}\group{\rejectset}\coloneqq\cset{\desirset\in\overline{\desirsets}_{\mathrm{M}}}{\rejectset\subseteq\rejectset[\desirset]}$.
\end{theorem}
\noindent
This representation result is akin to the one proved by Seidenfeld et al \cite{seidenfeld2010}, but without the additional condition of weak Archimedeanity they impose.
In order to better explain this, and to provide this result with some extra intuition, we take a closer look at the mixing sets of desirable options that make up our representation.
The following result is an equivalent characterisation of such sets.

\begin{proposition}\label{prop:mixingequalslexicographic}
Consider any set of desirable options $\desirset\in\overline{\desirsets}$ and let $\desirset^\mathrm{c}\coloneqq\opts\setminus\desirset$.
Then $\desirset$ is mixing if and only if\/ $\posi(\desirset^\mathrm{c})=\desirset^\mathrm{c}$.
\end{proposition}
\noindent
So we see that the coherent sets of desirable options that are also mixing are precisely those whose complement is again a convex cone.\footnote{Recall that coherent sets of desirable options are convex cones because of Axiom~\ref{ax:desirs:cone}.}
They are therefore identical to the \emph{lexicographic} sets of desirable options sets introduced by Van Camp et al.~\cite{2017vancamp:phdthesis,2018vancamp:lexicographic}.
What makes this particularly relevant and interesting is that these authors have shown that when $\opts$ is the set of all gambles on some finite set~$\mathcal{X}$ and $\opts_{\optgt0}=\cset{\opt\in\opts}{\opt\geq0\text{ and }\opt\neq0}$, then the sets of desirable options in $\overline{\desirsets}$ that are lexicographic---and therefore mixing---are exactly the ones that are representable by some lexicographic probability system that has no non-trivial Savage-null events.
This is, of course, the reason why they decided to call such coherent sets of desirable options \emph{lexicographic}.
Because of this connection, it follows that in their setting, Theorem~\ref{theo:mixingrepresentation:twosided} implies that every mixing choice model can be represented by a set of lexicographic probability systems.

Due to the equivalence between coherent lexicographic sets of desirable options and mixing ones on the one hand, and between total sets of desirable options and maximal coherent ones on the other, the following proposition is an immediate consequence of a similar result by Van Camp et al.~\cite{2017vancamp:phdthesis,2018vancamp:lexicographic}.
It shows that the total sets of desirable options constitute a subclass of the mixing ones: mixingness is a weaker requirement than totality.

\begin{proposition}\label{prop:mixingiftotal}
Every total set of desirable options is mixing: $\overline{\desirsets}_{\mathrm{T}}\subseteq\overline{\desirsets}_{\mathrm{M}}$.
\end{proposition}
\noindent
By combining this result with Theorems~\ref{theo:totalrepresentation:twosided} and~\ref{theo:mixingrepresentation:twosided}, it follows that every total set of desirable options sets is mixing, and similarly for rejection and choice functions.
So mixingness is implied by totality for non-binary choice models as well.
Since totality is arguably the more intuitive of the two, one might therefore be inclined to discard the mixing property in favour of totality.
We have nevertheless studied the mixing property in some detail, because it can be combined with other properties, such as the notions of Archimedeanity studied in the next section.
As we will see, this combination leads to very intuitive representation, where the role of lexicographic probability systems is taken over by expectation operators---called linear previsions.

\section{Imposing Archimedeanity}\label{sec:archimedeanity}

There are a number of ways a notion of Archimedeanity can be introduced for preference relations and choice models \cite{aumann1962,aumann1964,seidenfeld1995,seidenfeld2010,nau2006}.
Its aim is always to guarantee that the real number system is expressive enough, or more precisely, that the preferences expressed by the models can be represented by (sets of) \emph{real-valued} probabilities and utilities, rather than, say, probabilities and utilities expressed using hyper-reals.
Here, we consider a notion of Archimedeanity that is close in spirit to an idea explored by Walley \cite{walley1991,walley2000} in his discussion of so-called \emph{strict desirability}.

For the sake of simplicity, we will restrict ourselves to a particular case of our abstract framework,\footnote{It is possible to introduce a version of our notion of Archimedeanity in our general framework as well, but explaining how this works would take up much more space than we are allowed in this conference paper.} where $\opts\coloneqq\mathcal{L}(\mathcal{X})$ is the set of all gambles on a set of states $\mathcal{X}$ and $\opts_{\optgt0}=\gbls_{\mathrm{sp}}(\mathcal{X})\coloneqq\cset{\opt\in\mathcal{L}(\mathcal{X})}{\inf\opt>0}$.
We identify every real number $\mu\in\mathbb{R}$ with the constant gamble that takes the value $\mu$, and then define Archimedeanity as follows.

\begin{definition}[Archimedean set of desirable options]\label{def:gamblearchimedean:sets}
We call a set of desirable options $\desirset\in\mathbf{D}$ \emph{Archimedean} if it is coherent and satisfies the following openness condition:
\begin{enumerate}[label=$\mathrm{D}_{\mathrm{A}}$.,ref=$\mathrm{D}_{\mathrm{A}}$,leftmargin=*]
\item\label{ax:desirs:gamblearch} for all $\opt\in\desirset$, there is some $\epsilon\in\reals_{>0}$ such that $\opt-\epsilon\in\desirset$.
\end{enumerate}
We denote the set of all Archimedean sets of desirable options by $\overline{\desirsets}_{\mathrm{A}}$, and let $\overline{\desirsets}_{\mathrm{M,A}}$ be the set of all Archimedean sets of desirable options that are also mixing.
\end{definition}

What makes Archimedean and mixing Archimedean sets of desirable options particularly interesting, is that they are in a one-to-one correspondence with coherent lower previsions and linear previsions \cite{walley1991}, respectively.

\begin{definition}[Coherent lower prevision and linear prevision]\label{def:cohlp}
A \emph{coherent lower prevision} $\lowprev$ on $\gblson{\mathcal{X}}$ is a real-valued map on $\mathcal{L}(\mathcal{X})$ that satisfies
\begin{enumerate}[label=$\mathrm{LP}_{\arabic*}$.,ref=$\mathrm{LP}_{\arabic*}$,leftmargin=*]
\item\label{ax:lowprev:inf} $\lowprev(\opt)\geq\inf\opt$ for all $\opt\in\gblson{\mathcal{X}}$;
\item\label{ax:lowprev:lambda} $\lowprev(\lambda\opt)=\lambda\lowprev(\opt)$ for all $\opt\in\gblson{\mathcal{X}}$ and $\lambda\in\reals_{>0}$;
\item\label{ax:lowprev:super} $\lowprev(\opt+\altopt)\geq\lowprev(\opt)+\lowprev(\altopt)$ for all $\opt,\altopt\in\gblson{\mathcal{X}}$;
\end{enumerate}
A \emph{linear prevision} $\linprev$ on $\gblson{\mathcal{X}}$ is a coherent lower prevision that additionally satisfies
\begin{enumerate}[label=$\mathrm{P}_{\arabic*}$.,ref=$\mathrm{P}_{\arabic*}$,leftmargin=*,start=3]
\item\label{ax:linprev:additive}$\linprev(\opt+\altopt)=\linprev(\opt)+\linprev(\altopt)$ for all $\opt,\altopt\in\gblson{\mathcal{X}}$;
\end{enumerate}
We denote the set of all coherent lower previsions on $\gblson{\mathcal{X}}$ by $\underline{\mathbf{P}}$ and let $\mathbf{P}$ be the subset of all linear previsions.
\end{definition}

In order to make the above-mentioned one-to-one correspondences explicit, we introduce the following maps.
With any set of desirable options $\desirset$ in $\mathbf{D}$, we associate a (possibly extended) real functional $\lowprev[\desirset]$ on $\gblson{\mathcal{X}}$, defined by
\begin{equation}\label{eq:lpfromdesir}
\lowprev[\desirset](\opt)\coloneqq\sup\cset{\mu\in\mathbb{R}}{\opt-\mu\in\desirset},
\text{ for all $\opt\in\gblson{\mathcal{X}}$}.
\end{equation}
Conversely, with any (possibly extended) real functional $\lowprev$ on $\gblson{\mathcal{X}}$, we associate a set of desirable options
\begin{equation}\label{eq:desirfromlp}
\desirset[\lowprev]\coloneqq\cset{\opt\in\gblson{\mathcal{X}}}{\lowprev(\opt)>0}.
\end{equation}
Our next result shows that these two maps lead to an isomorphism between $\overline{\desirsets}_{\mathrm{A}}$ and $\underline{\mathbf{P}}$, and similarly for $\overline{\desirsets}_{\mathrm{M,A}}$ and $\mathbf{P}$.

\begin{proposition}\label{prop:isomorphismsmarch}
For any Archimedean set of desirable options $\desirset$, $\lowprev[\desirset]$ is a coherent lower prevision on $\mathcal{L}(\mathcal{X})$ and $\desirset[{\lowprev[\desirset]}]=\desirset$.
If $\desirset$ is moreover mixing, then $\lowprev[\desirset]$ is a linear prevision.
Conversely, for any coherent lower prevision $\lowprev$ on $\mathcal{L}(\mathcal{X})$, $\desirset[\lowprev]$ is an Archimedean set of desirable options and $\lowprev[{\desirset[\lowprev]}]=\lowprev$.
If $\lowprev$ is furthermore a linear prevision, then $\desirset[\lowprev]$ is mixing.
\end{proposition}

The import of these correspondences is that any representation in terms of sets of Archimedean (mixing) sets of desirable options is effectively a representation in terms of sets of coherent lower (or linear) previsions.
As we will see, these kinds of representations can be obtained for sets of desirable option sets---and hence also rejection and choice functions---that are themselves Archimedean in the following sense.

\begin{definition}[Archimedean set of desirable option sets]\label{def:gamblearchimedean:setsofsets}
We call a set of desirable option sets $\rejectset\in\mathbf{K}$ \emph{Archimedean} if it is coherent and satisfies
\begin{enumerate}[label=$\mathrm{K}_{\mathrm{A}}$.,ref=$\mathrm{K}_{\mathrm{A}}$,leftmargin=*]
\item\label{ax:rejects:gamblearch} for all $\optset\in\rejectset$, there is some $\epsilon\in\reals_{>0}$ such that $\optset-\epsilon\in\rejectset$.
\end{enumerate}
We denote the set of all Archimedean sets of desirable option sets by $\overline{\rejectsets}_{\mathrm{A}}$, and let $\overline{\rejectsets}_{\mathrm{M,A}}$ be the set of all Archimedean sets of desirable options that are also mixing.
\end{definition}
\noindent
This notion easily translates from sets of desirable option sets to rejection functions.

\begin{proposition}
\label{prop:gamblearchimedeanityforrejectionfunctions}
Consider any set of desirable option sets~$\rejectset\in\mathbf{K}$ and any rejection function~$\rejectfun$ that are connected by Equation~\eqref{eq:interpretation:rejectfuns:after:irreflexivity:and:additivity:intermsof:K:Gert}.
Then $\rejectset$ is Archimedean if and only if $\rejectfun$ is coherent and satisfies
\begin{enumerate}[label=$\mathrm{R}_{\mathrm{A}}$.,ref=$\mathrm{R}_{\mathrm{A}}$,leftmargin=*]
\item\label{ax:rejectfun:gamblearch} for all $\optset\in\mathcal{Q}$ and $\opt\in\opts$ such that $\opt\in\rejectfun(\optset\cup\set{\opt})$, there is some $\epsilon\in\reals_{>0}$ such that $\opt\in\rejectfun((\optset-\epsilon)\cup\set{\opt})$.
\end{enumerate}
\end{proposition}

A first and basic result is that our notion of Archimedeanity for sets of desirable option sets is compatible with that for sets of desirable options.

\begin{proposition}\label{prop:gamblearchbinaryiffD}
For any set of desirable options $\desirset\in\mathbf{D}$, $\rejectset[\desirset]$ is Archimedean (and mixing) if and only if~$\desirset$ is, so $\rejectset[\desirset]\in\overline{\rejectsets}_{\mathrm{A}}\Leftrightarrow\desirset\in\overline{\desirsets}_{\mathrm{A}}$ and $\rejectset[\desirset]\in\overline{\rejectsets}_{\mathrm{M,A}}\Leftrightarrow\desirset\in\overline{\desirsets}_{\mathrm{M,A}}$.
\end{proposition}

In order to state our representation results for Archimedean choice models that are not necessarily binary, we require a final piece of machinery: a topology on $\overline{\desirsets}_{\mathrm{A}}$ and $\overline{\desirsets}_{\mathrm{M,A}}$, or equivalently, a notion of closedness.
We do this by defining the convergence of a sequence of Archimedean sets of desirable options $\set{\desirset[n]}_{n\in\mathbb{N}}$ in terms of the point-wise convergence of the corresponding sequence of coherent lower previsions:
\begin{equation}\label{eq:archimedeanlimit}
\lim_{n\to+\infty}\desirset[n]=\desirset
\Leftrightarrow
\group[\big]{\forall\opt\in\gblson{\mathcal{X}}}\lim_{n\to+\infty}\lowprev[{\desirset[n]}](\opt)=\lowprev[\desirset](\opt).
\end{equation}
A set $\mathcal{D}\subseteq\overline{\desirsets}_{\mathrm{A}}$ of Archimedean sets of desirable options is then called \emph{closed} if it contains all of its limit points, or equivalently, if the corresponding set of coherent lower previsions---or linear previsions when $\mathcal{D}\subseteq\overline{\desirsets}_{\mathrm{M,A}}$---is closed with respect to point-wise convergence.

Our final representation results state that a set of desirable option sets $\rejectset$ is Archimedean if and only if it can be represented by such a closed set, and if $\rejectset$ is moreover mixing, the elements of the representing closed set are as well.

\begin{theorem}[Representation for Archimedean choice functions]\label{theo:rejectsets:representation:gamblearch:twosided}
A set of desirable option sets $\rejectset\in\mathbf{K}$ is Archimedean if and only if there is some non-empty closed set $\mathcal{D}\subseteq\overline{\desirsets}_{\mathrm{A}}$ of Archimedean sets of desirable options such that $\rejectset=\bigcap\cset{\rejectset[\desirset]}{\desirset\in\mathcal{D}}$.
The largest such set~$\mathcal{D}$ is then\/ $\overline{\desirsets}_{\mathrm{A}}(\rejectset)\coloneqq\cset{\desirset\in\overline{\desirsets}_{\mathrm{A}}}{\rejectset\subseteq\rejectset[\desirset]}$.
\end{theorem}

\begin{theorem}[Representation for Archimedean mixing choice functions]\label{theo:convex:rejectsets:representation:gamblearch:twosided}
A set of desirable option sets $\rejectset\in\mathbf{K}$ is mixing and Archimedean if and only if there is some non-empty closed set $\mathcal{D}\subseteq\overline{\desirsets}_{\mathrm{M,A}}$ of mixing and Archimedean sets of desirable options such that $\rejectset=\bigcap\cset{\rejectset[\desirset]}{\desirset\in\mathcal{D}}$.
The largest such set~$\mathcal{D}$ is then\/ $\overline{\desirsets}_{\mathrm{M,A}}(\rejectset)\coloneqq\cset{\desirset\in\overline{\desirsets}_{\mathrm{M,A}}}{\rejectset\subseteq\rejectset[\desirset]}$.
\end{theorem}

If we combine Theorem~\ref{theo:convex:rejectsets:representation:gamblearch:twosided} with the correspondence result of Proposition~\ref{prop:isomorphismsmarch}, we see that Axioms~\ref{ax:rejects:removezero}--\ref{ax:rejects:mono} together with~\ref{ax:rejects:removepositivecombinations} and~\ref{ax:rejects:gamblearch} characterise exactly those choice models that are based on E-admissibility with respect to a closed---but not necessarily convex---set of linear previsions.
In much the same way, Theorem~\ref{theo:rejectsets:representation:gamblearch:twosided} can be seen to characterise a generalised notion of E-admissibility, where the representing objects are coherent lower previsions.
Walley--Sen maximality \cite{troffaes2007,walley1991} can be regarded as a special case of this generalised notion, where only a single representing coherent lower prevision is needed.

\section{Conclusion}\label{sec:conclusion}

The main conclusion of this paper is that the language of desirability \emph{is} capable of representing non-binary choice models, provided we extend it with a notion of disjunction, allowing statements such as `at least one of these two options is desirable'.
When we do so, the resulting framework of sets of desirable options turns out to be a very flexible and elegant tool for representing set-valued choice.
Not only does it include E-admissibility and maximality, it also opens up a range of other types of choice functions that have so far received little to no attention.
All of these can be represented in terms of sets of strict preference orders or---if additional properties are imposed---in terms of sets of strict total orders, sets of lexicographic probability systems, sets of coherent lower previsions or sets of linear previsions.

Another important conclusion is that our axiomatisation for general (possibly non-binary) choice models allows for representations in terms of `atomic' models, which in themselves represent binary choice.
However, this should of course not be taken to mean that our choice models are essentially binary.
Indeed, it follows readily from our representation theorems that the binary aspects $\desirset[\rejectset]$ of a non-binary choice model $\rejectset$ are captured by the intersections of the representing sets of desirable options, but the representation is much more powerful than that, because it also extends to the non-binary aspects of choice.

This distinction between the binary level and the non-binary one also leads us to the following important words of caution, which are akin to an earlier observation made by Quaeghebeur \cite{quaeghebeur2015:partial:partial}.
At the binary level, choice is represented by a set of desirable options, which can---under certain assumptions such as Archimedeanity---be identified with a convex closed set of linear previsions.
We have also seen in Theorem~\ref{theo:convex:rejectsets:representation:gamblearch:twosided} that the (binary \emph{and} non-binary) aspects of mixing and Archimedean choice can be fully represented by a closed set of mixing and Archimedean sets of desirable options, each of which is, by Proposition~\ref{prop:isomorphismsmarch}, equivalent to a linear prevision.
So, in this case there is a representation in terms of a set of linear previsions both at the binary level and at the general (binary \emph{and} non-binary) level, but these two sets of linear previsions will typically be different, and they play a very different role.
To put it very bluntly: sets of linear previsions à la Walley \cite{walley1991} should not be confused with sets of linear previsions---credal sets---à la Levi \cite{levi1980a}.

To conclude, what we have done here, in a very specific sense, is to introduce a way of dealing with statements of the type `there is some option in the option set~$A$ that is strictly preferred to~$0$'.
Axioms such as \ref{ax:rejects:removezero}--\ref{ax:rejects:mono} can then be seen as the logical axioms---for deriving new statements from old---that govern this language.
Our representation theorems provide a \emph{semantics} for this language in terms of desirability, and they show that the corresponding logical system is sound and complete.

In our future work on this topic, we intend to investigate how we can let go of the closedness condition in Theorems~\ref{theo:rejectsets:representation:gamblearch:twosided} and~\ref{theo:convex:rejectsets:representation:gamblearch:twosided}.
We expect to have to turn to other types of Archimedeanity; variations on Seidenfeld et al.'s weak Archimedeanity \cite{seidenfeld2010,cozman2018:evenlyconvex} come to mind.
We also intend to show in more detail how the existing work on choice models for horse lotteries \cite{seidenfeld2010} fits nicely within our more general and abstract framework of choice on linear option spaces.
And finally, we intend to further develop conservative inference methods for coherent choice functions, by extending our earlier natural extension results~\cite{debock2018} to the more general setting that we have considered here.

\section*{Acknowledgements}
This work owes a large intellectual debt to Teddy Seidenfeld, who introduced us to the topic of choice functions.
His insistence that we ought to pay more attention to non-binary choice if we wanted to take imprecise probabilities seriously, is what eventually led to this work.

The discussion in Arthur Van Camp's PhD~thesis \cite{2017vancamp:phdthesis} was the direct inspiration for our work here, and we would like to thank Arthur for providing a pair of strong shoulders to stand on.

As with most of our joint work, there is no telling, after a while, which of us two had what idea, or did what, exactly.
We have both contributed equally to this paper.
But since a paper must have a first author, we decided it should be the one who took the first significant steps: Jasper, in this case.

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