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An Ashton P. Woods Publication
#ITLQBM - Blackness & Equality Under Attack (Video Update)
Intersectionality Through the Lens of a Queer Black Man
Waller, TX
Houston City Council
In recent posts, I have discussed at length the Sandra Bland case and on a macro level about how #BlackLivesMatter. On the flip side of the coin, my involvement with the #HERO (Houston equal rights ordinance) has been very active, especially since a rushed ruling just came down from the Texas Supreme Court. These are the intersections that I and many of my counterparts of Color speak of when we are doing the work to demand respect and the decriminalization of Blackness. As of today, I am faced with a choice to testify at Houston City Hall tomorrow during the Council meeting on behalf of HERO or be in Waller county to meet with my cohorts in front of the county jail to stand in solidarity for our fallen sister Sandra and discuss next steps beyond our planned event on August 9th.
***For some background on both issues follow these links:
http://ashtonpwoods.strengthinnumbershouston.com/2015/07/a-call-to-action-blacklivesmatter.html
http://ashtonpwoods.strengthinnumbershouston.com/2015/07/hero-interrupted-texas-supreme-court.html
http://ashtonpwoods.strengthinnumbershouston.com/2015/07/what-happened-to-sandra-bland-we-need.html
When you think about it HERO intersects with the movement for Black lives in ways that are quite interesting. Here is how:
*The Houston Equal Rights Ordinance provides protections in employment (both public and private), housing and public accommodations for the following classes as it applies to my intersections:
- sex - race - sexual orientation - color- ethnicity- gender identity*
As a Black man I choose to be part of movements that impact all People of Color for the better and isn't it interesting that regardless of where I stand, I am under attack by the current power structure. In this case the power structure is clergy, government, law enforcement and at its foundation is homophobia, racism, prejudice and bigotry. In a previous post I said "...Honestly, the factors that I have just pointed out are a primer for the real discussion about being a "Black Queer Man." Take all of those factors and apply which ever resonates to you individually now add to it being out and Gay. We still have to deal with those stigmas that our heterosexual counterparts have to deal with and then some. We are attacked by those who look like us for not cosigning what I described earlier as what it means to be a strong Black man, we are told that because of who we are that we do not exist. IN FACT, we do have the same issues and then we are treated as if we are no longer Black and are still niggers in the eyes of society on a systemic and structural level. Now, let me take that down to a micro level and deal with all that I previously mentioned and combine it with the issues that we experience as Queer Black Men of color. First, we still have to deal with racism from a group that we are supposedly part of and are "welcome" to, which is the mainstream LGBT community. In many ways we are shut out by mainstream White gays unless we bring something to the table that they just cannot function without, DIVERSITY...We are only needed when it's beneficial and then we have to live in our poverty that people think we don't experience. We work in jobs that not only attack our race systemically and structurally, now we have the vector of homophobia in the work place to deal with. In fact, the structural and systematic homophobia that plague the LGBT community at large is much worse on people of color due to the same racial barriers to access as our heterosexual counterparts. We work in low paying jobs that range from fast food to hospitality and then we have to deal with being looked down upon by the ones that look like us and are gay like us who are making some good pay and etc. There is an overriding theme that we are being attacked from all sides, do you see it?"
When I wrote this, I had no clue that it would be so relevant to my feelings and what I am currently experiencing. I feel extremely stressed out and emphasis on being under attack. Do you know what it feels like to have people vying to be Mayor of Houston, currently elected/appointed judges tell me and so many others that an ordinance that protects me as an individual in six different ways is void in the eyes of the state of Texas. It is a constant feeling of siege when I hear about the loss of a Black life at the hands of some cop who likes to use excessive force and abuses his/her power. There is no way that Sandra Bland killed herself and it is clear that this is a cover up of epic proportions. Black Lives truly do matter and intersectionally speaking, this is the time to rise up, not just for the protections from discrimination and for law enforcement to do right by us. The time is now to rise up and recognize that we are being pushed out of affordable housing while being moved to the outskirts of the city, we lack access good healthcare facilities, and our kids are sent to schools in one of the most segregated school districts in the country that has a focus on the early criminalization of students. Who expels a kindergartner? WE are under a level of attack that cant be quantified or measured, we need our voting rights restored, we need affirmative action to be protected! What good is it if I can get same sex married at city hall and walk out to the possibility of some bigot protesting my rights, a cop ready to put his bullet in my brain, the prospect of losing my job for being gay, and a host of other issues not being discussed. Lets expose the hate, racism, and bigotry where ever it may be hiding....
Posted by Ashton P. Woods at 1:31:00 PM No comments:
Labels: #HERO, #ITLQBM
A CALL TO ACTION: #BlackLivesMatter
In remembrance of Michael Brown who died last year (8/9/14) from an encounter with violent police officer Darren Wilson and Sandra Bland who Died on July 13th under very suspicious circumstances an event will be held at Waller County Jail. This is a response that requires the participation of all who demand justice in the face of police brutality, racism and hate. Our blackness is not a crime, we are not terrorists, and we will not lie down and take this abuse anymore!
Waller County Sheriff's Department
1.0108 Google reviews
Address: 701 Calvit St, Hempstead, TX 77445
THIS IS A CALL TO ACTION: ALL TEXAS ACTIVISTS ARE INVITED, ALL ACTIVISTS FROM AROUND THE USA ARE INVITED.
#HERO Interrupted - Texas Supreme Court Ruling
Yesterday the Texas Supreme Court ruled in favor of the opponents of the Houston Equal Rights Ordinance. This comes as Dave Wilson's law suit to change the city charter with his rejected petition that was signed and supported by Mayoral candidate Ben Hall. You can read what I wrote about from that angle HERE and HERE. In its ruling the court ordered the Houston City Council to repeal an LGBT nondiscrimination ordinance it passed in 2014 or put a referendum opponents had sought on this November's ballot.
"Once the City Council received the City Secretary's certification, it had a ministerial duty to act," the court stated. "If the City Council does not repeal the ordinance by August 24, 2015, then by that date the City Council must order that the ordinance be put to popular vote during the November 2015 election."
Here is Texas Supreme court ruling: http://www.txcourts.gov/media/1047360/140667.pdf
Protect HERO Meeting
Tomorrow at 12:30pm
Resurrection Metropolitan Community Church in Houston, TexasBrad Pritchett invited you to HOUequality's event
HOUequality is an information clearinghouse regarding the Houston Equal Rights Ordinance
HOUEQUALITY.NATIONBUILDER.COM
Here is some background on HERO:
During the first half of 2014 the effort to passed HERO was under way and it was met with delays, protests and support. The ordinance on the table would protect 15 classes of citizens and give a way to affordably challenge those who commit discrimination.
*Houston Equal Rights Ordinance provides protections in employment (both public and private), housing and public accommodations for the following classes:
- sex - age - disability
- race - familial status - sexual orientation
- color - marital status - genetic information
- ethnicity - military status - gender identity
- national origin - religion - pregnancy
Here is the Ordinance:
Since its passage in an 11-6 vote of the City Council on May 28, 2014:
In July, after thirty days of collecting signatures, opponents of the ordinance turned in a petition that they claimed to be more than the minimum number of signatures needed to trigger a November 2014 vote on whether to repeal the measure. Once the city received the petition, the City Secretary's office had 30 days to verify the alleged 50,000 signatures that HERO opponents collected. The required minimum threshold to beat HERO was 17,269 signatures from registered Houston voters. The issue that opponents attacked the ordinance for, boiled down to the protections it extends to Lesbian, Gay, Bisexual and mainly transgender residents.
They basically used the "the man dressed as a woman to rape women and children" argument which is disproven by this:
Houston Penal Code: Sec. 28-20. Entering restrooms of opposite sex.permanent link to this piece of content It shall be unlawful for any person to knowingly and intentionally enter any public restroom designated for the exclusive use of the sex opposite to such person's sex without the permission of the owner, tenant, manager, lessee or other person in charge of the premises, in a manner calculated to cause a disturbance. (Code 1968, § 28-42.6; Ord. No. 72-904, § 2, 6-2-72)
The community got together to do an independent count and verification of the signatures and by or count, there was fraud, forgeries and etc. Then in the first week of August, in compliance with the city rules, Mayor Annise Parker and then City Attorney Dave Feldman released their findings on the petition count. As I was at the press conference here is what Feldman had to say: "There are simply too many documents with irregularities and problems to overlook. The petition is simply invalid." With that statement the signatures were declared to be invalidated:
We expected a court battle and got one, the lawsuit was immediately filed:
In between the filing and the final ruling this is what happened:
In this trail there was a jury verdict and a final ruling by judge Schaffer:
2014-44974 HERO Trial Verdict
Herotrial Petition Ruling by Brad Pritchett
The rest is history....
Labels: #HERO
Dave Wilson - Your Campaign of HATE Will Not Win: Enter Ben Hall 4 Mayor
What many of you may not know, is that for the past few months Wilson has been making attempts at gathering signatures to attack the TRANSGENDER community:
"PETITION FOR A CHARTER AMENDMENT TO PROHIBIT MEN (WHO PERCEIVE OR EXPRESS THEMSELVES AS WOMEN) FROM USING WOMEN'S SEX - SEGREGATED FACILITIES OR WOMEN (WHO PERCEIVE OR EXPRESS THEMSELVES AS MEN) FROM USING THE MEN'S SEX - SEGREGATED FACILITIES."
Now look closer at the petition, which he mailed out to Houstonians:
- He already has their voter registration info on the form with a bar code.
- All they have to do is sign an date
This petitions for a charter amendment that would require the city to change all policy so that gender is interpreted as the gender a person was assigned at birth. It would also require "all entities doing business with the city to abide by the same definition."
******THIS IS AN ALL CALL TO THE LGBT COMMUNITY TO TAKE A STAND AGAINST DAVE WILSON, SHOULD THE CITY OF HOUSTON ACCEPT THIS PETITION *******
I want you to know that, regardless of how this scenario goes, this is a warning to the community at large that we need to be ready for challenges to our rights. Yes, we now have HERO and even the right to marry, but Wilson and his friends will try to chip away at us and we need to be ready.
**********************UPDATE************************UPDATE*****************************UPDATE**********************
Not long after making the original post above, THIS HAPPENED:
Then today (7/10/15): "After the city rejecting his petitions, Dave Wilson has filed a lawsuit against the City of Houston.
So yes, another lawsuit with an end goal that would damage our city's reputation.
It should be noted that Dave Wilson is a current Houston Community College Trustee, but rather than gaining notoriety doing his elected job, his quest to deny LGBT people equality is what he has become famous for." - HOUequality
HERE IS THE LAW SUIT:
Wilson Lawsuit by Brad Pritchett
ENTER HOUSTON 2015 MAYORAL CANDIDATE BEN HALL
While I was in Phoenix, AZ it was reported by a couple of organizations HOUequality & Houston Stonewall Young Dems that none other than Ben Hall Signed the petitions that my last post talked about:
Now at this point I have always made it known that a vote for Ben Hall is a vote against yourself! He released a statement on his website stating that he signed the petition:
Ben Hall you are a BIGOT and you need to come to terms with that and soon, these are not "false accusations" that we are aiming at you. You said, "I proudly signed the petition" which means that you are continuing the disproved and idiotic assertion that the Houston Equal Rights Ordinance allows a "male sexual predator to simply dress up as a woman to gain access to a woman's restroom." You know that what you are saying is baseless and is a straight up lie used evoke an irrational fear of transgender Houstonians.
Houston Penal Code: Sec. 28-20. Entering restrooms of opposite sex.permanent link to this piece of contentIt shall be unlawful for any person to knowingly and intentionally enter any public restroom designated for the exclusive use of the sex opposite to such person's sex without the permission of the owner, tenant, manager, lessee or other person in charge of the premises, in a manner calculated to cause a disturbance. (Code 1968, § 28-42.6; Ord. No. 72-904, § 2, 6-2-72)
The Houston Equal Rights Ordinance provides protections in employment (both public and private), housing and public accommodations for the following classes:
I ask you to reconsider if you plan on voting for Ben Hall, He is not the right person to lead Houston and for more thoughts please check out what Monica Roberts had to say on Transgriot...
What Happened to Sandra Bland??? We Need Answers (video)
#SayHerName
Posted by Ashton P. Woods at 3:45:00 AM No comments:
Dear Presidential Candidates: Do Black Lives Really Matter?
Update: As we approach 2016, much has changed over the last few months since this open letter was originally posted. My support for any presidential candidate has been suspended since October, due to the following reasons:
1. I vote my issues and a candidate can say they value and will address my issues, issues that plague the Black community, but I don't trust the acknowledgement of the issue as a means of pacification. Too often candidates say what they think people want to hear and the truth is that I just have a healthy level of distrust.
2. Houston just completed it's 2015 cycle of municipal, county, and state elections that just so happened to be more important than 2016. We were at risk of becoming a city run by conservative interests wanting to drag Houston back 50 years and home needed to be taken care of.
3. Now that I/we can focus on 2016, home must be the focus once again. Why? There are local judicial seats, district atty and other races outside of our Presidential election. We have work to do on all levels and each level just so happens to have an effect on how our Black bodies are treated. Keep your eyes open!
***WE WON THE BATTLE IN 2015, BUT WE MUST PREPARE FOR WAR IN 2016***
As a Black man who currently is throwing his support behind Hillary Clinton, I am not going to allow that to deter me from the cause for Black Lives to Matter. On this past Saturday, I was part of a group that planned and carried out a civil disruption at the #NN15 Presidential Forum, which was led by women. The reason it was led by Women is because it is now the moment to not only center Blackness, but to center Black Women in the context that they experience the same brutality and structural racism that Black Men do. The event gave birth to the question "will you say her name?" for the candidates and "do Black lives really matter?' Now as the days have progressed and we wait for the truth to come out about Sandra Bland, people have come to the overriding conclusion that we activist just wanted the candidates to say something. What we really want is to know WHAT THE CANDIDATES WILL DO to combat racism in its institutional, systemic, and its structural forms with in the confines of the Executive powers. Yes that is great that a speech with a populist tone can warm the hearts of thousands of men and women around this nation, but the truth is that many of us are NOT SOLD on what is being said. We want to know what you candidates are doing RIGHT NOW from your current position and how that looks with you as the President of these United States. The thing is that Many of the people who organized this recent disruption are not only Black, but are Queer and other shades of Brown. This means that we are stuck at a cross roads of experiences that includes, Homophobia, BIphobia, Transphobia and a host of other issues in communities of orientation as well as from those out side of aforementioned communities.
We are hurting in communities of color, it is nice that the Justice Dept. is doing work to change a lot of the institutionally racist structure on the on the federal level at the directive of President Obama. We more than that, many of you candidates hold federal office and frankly you hold more power at the moment to pass legislation that will reverse many of the ills that we experience as mentioned earlier. I call bullshit until further notice, put your money where your policy mouth is and take concrete actions right now! Not just with body cameras and reducing military equipment rations to police forces. How about you stop the privatization of the education system, review penal codes, and other mandates within your reach of control that other departments cannot change with an act of Congress. So tell us what is your plan, solution, or idea to combat these issues?
#ITLQBM - Netroots Presidential Forum Interrupted #SayHerName #BlackLivesMatter
On Saturday, July 18 at 11:07 a.m., I and many Black and Brown #NN15 attendees disrupted the Presidential forum as Mike O'Malley and Bernie Sanders were in a Q & A session about why they should be the next President. I dedicate my part in organizing this action to Sandra Bland and many of our Queens lost do to forms of police violence against Blackness. The following video is from my point of view, so sit back and watch democracy in action #BlackLivesMatter #SayHerName #IfIDieInPoliceCustody:
*This is one the FIRST (maybe the first) disruptions where Sandra Bland was honored nationally along with so many of the other powerful Black women (Cis/Trans) lost to police brutality.
Labels: #ITLQBM, #NN15 #LGBTNN15 #QPOCNN15
Ashton Goes to Phoenix #NN15/#ITLQBM: EQUALITY ACTION CAUCUS (UPDATE)
After the Black Caucus, I went to the Equality Action Caucus where activists and leaders of the broad based equality to discuss key priorities and connect with allies. We attempted talk about the progress and future directions of the equality. The central question of the event was "where do we go from here?" When I got a chance to speak from a place of Blackness, support of the Transgender community and other marginalized groups within the SGL/LGBT community. I gave a disclaimer because of a gut feeling that I had, that my comments would come off as angry and abrasive. I stated that I had three points that would point out problematic language, Trans erasure and Black erasure. I was interrupted several times by a White guy who happened to one one the moderators. I kept to my principles and made my points all while telling him to check his privileges and to stop interrupting me. This caused a very strong reaction from other people in attendance in my favor, OF ALL RACES. They even called him out on it and how he was making the space hostile and unwelcoming to folks of color and allies. Many people including myself left afterwards and it was very interesting and crazy how it all happened! Next year look for me to submit a proposal to these conferences to create Trans inclusive, safe for Blackness, SGL_LGBT spaces!
Labels: #ITLQBM, #LGBTNN15, #NN15, #QPOCNN15
Ashton Goes to Phoenix #NN15
For a week or so I will be in Phoenix, AZ for Netroots Nation 2015 to interact with other activists, politicians, and like minded individuals. This post will contain daily updates and photos from my time in Phoenix at the conference. For this first entry I would like to thank the Blue Nation Review and Netroots Connect for their extending scholarships to me in order for me to attend this week's events. I would like to thank all of my friends and supporters who made donations to help cover expenses that the scholarship did not cover. If you would like to make a donation please send it via PayPal to email address ashtonpwoods@gmail.com.
Netroots LGBT Connect (7/15)
During the Connect event I had the pleasure to meet so many like minded people, to learn, relearn, unlearn, and affirm a lot of things about how I practice my activism. We had an all day discussion on intersectionality, which was rather refreshing in so many ways. It was refreshing because the discussion is rarely held in my home town of Houston and it makes me a little annoyed (I will get over it). It is always the case that I leave home and find my place within a group of like minded people and I can truly be myself with out the subtext of respectability politics and the my way is better than your way attitudes. The were some moments of internal contention for me when I realized that I made sure to illustrate that it is bad to question a person's Blackness and therefore diminishing that person's experiences as a Black person. On the flip side of that coin, it was very much so annoying that there were some folks of color who just could not wash off the obvious Whitewash that they were wearing in a space with not very many Black people. It is what it is, and all that means is that I shall continue to stay in my lane and do the work that I am called to do, over all it was a great day of pre- Netroots activities....
Netroots Day One (7/16)
Today is the first day of Netroots and so far I have seen a hand full of Black people in a sea of White, there are many factors that contribute to this. I will not discuss these factors, YET, but I will say that this event has been very good for me to observe as far as my presence in this space. I went and registered for the weekend events.....
The first session that I attended was State Battlegrounds: The Real Fight for Voting rights where we discussed the passage of voting restrictions in nearly two dozen states since 2010. There were many nuances that came to light that ranged from actual voter IDs to Native American and mainstream voter suppression as a whole. I enjoyed the time that I had to listen to and even speak on the issue of voter suppression. Right now I am currently at We One: Overcoming Shared Opposition of the LGBT & Reproductive Rights Movements. At this event we are discussing the intersections of reproductive rights, religion, the LGBT community, and how they all affect/effect each other .
I went to the the Queer People of Color Caucus which provided a safe space for people who identified with being within the Queer spectrum and as People of Color. One of the many orders of housekeeping that I came to enjoy greatly is that the moderators of the event asked that those who did not identify as Queer or as a Person of Color to leave the room. Too many times when a space is created for people it is not respected and this time it wasn't allowed, but it did lead to a blow up in the common area know as the "Town Square" that houses booths for some of the organizations in attendance. One of the White people who were asked to leave the space went off at the LGBT connect booth and it was said that several people witnessed it. Here are some photos including Jennicet Eva Gutiérrez's appearance:
After the Queer POC caucus, I went to the Black caucus and we discussed at length whether or not the black community is, should be or part of the Progressive movement. The event was live broadcast via internet show "This Week in Blackness" and of course the hosts like Elon James White and other new friends in the selfies and event photos below....
More updates in a coming post : )
Sign-on Statement in Response to Michael Johnson Sentencing - CNP
Organizational Sign-on Statement in Response to Michael Johnson Sentencing
July 15, 2015 counternarrativeprojectnational
Joint Statement on theSentencing of Michael L. Johnson
Counter Narrative Project/Positive Women's Network (PWN-USA)/HIV Prevention Justice Alliance/ National Center for Lesbian Rights
On Monday July 13, 2015, Michael L. Johnson was sentenced to 30½ years in prison (a concurrent sentence) after being convicted of "recklessly infecting a partner with HIV" and "recklessly exposing partners to the virus." We are outraged by this sentencing and Johnson's incarceration. This represents a failure of the justice system and a blatant manifestation of structural violence in the lives of Black gay men.
The State of Missouri was able to convict Michael Johnson without having to prove that he had any intent to infect his sexual partners nor demonstrate that he was in fact the person who transmitted HIV to his sexual partners. We are outraged by the criminalization, arrests and imprisonment of those prosecuted under HIV criminalization laws. We will continue to fight for Michael, to repeal HIV criminalization laws, to dismantle the Prison Industrial Complex, and to end the stigma and violence perpetrated upon people living with HIV by these laws. With this mission in mind, we are calling for the following:
The Right for People Living with HIV to choose if, when, and how they disclose
HIV disclosure is not safe under every circumstance. People with HIV may face risks ranging from loss of employment to personal humiliation, custody battles, and violence resulting from disclosure. In addition, the burden of proving disclosure rests on the person living with HIV, not her/his partner. While we are committed to helping create a world where disclosure of HIV status is safe, we reject the notion that disclosure of HIV status should be coerced by the State. Laws criminalizing alleged non-disclosure do not make it easier to disclose, and do not protect people from acquiring HIV.
An HIV prevention policy that relies on disclosure of HIV status fails to account for the fact that data shows a person is more likely to contract HIV from a sexual partner who is unaware of their HIV positive status and that effective care and treatment for people living with HIV reduces the likelihood of transmission to almost zero. The best approach for those who are HIV-negative or of unknown HIV status is to practice self-efficacy and care – an approach which could include prevention strategies such as: (1) Learning how HIV and other STDs are transmitted and effective ways to prevent contracting the virus (2) Taking PrEP (3) Using condoms (4) Getting tested with partners for HIV and other STDs (5) Engaging in lower risk sexual activities (6) Identifying support and resources to leave unhealthy relationships that don't support protecting oneself (7) Confronting insecurities that lead oneself to seek validation by engaging in higher risk sexual behavior.
Today, HIV is no longer a near certain death sentence. With timely diagnosis and proper treatment HIV has become a manageable chronic disease similar to diabetes. People living with HIV can and are living long, healthy, and wonderful lives. And yet, the stigma remains. The truth is that criminalization of HIV is not really about our fear of HIV itself but the stigma that is attached to it. Those of us who are not living with HIV fear that if we contract HIV that we will suffer a lifetime of discrimination and rejection. Given this fear, those of us who are HIV negative should understand why someone who is living with HIV would not disclose her or his HIV status. Therefore, the real target is HIV stigma, including institutionalized stigma which manifests in laws and policies such as HIV criminalization.
Advocacy Against HIV Criminalization is Advocacy Against Mass Incarceration
HIV is a human rights issue, and criminalization of people living with HIV is a social justice issue. Resisting the Prison Industrial Complex means understanding how inequities in the HIV epidemic and sentencing disparities within the criminal justice system interface with laws that criminalize people with HIV. HIV criminalization laws serve as a means of expanding the categories of people subject to imprisonment, by virtue of an immutable characteristic-positive HIV status. In effect, this creates a biological underclass. HIV criminalization does not provide solutions nor will throwing people into prison lower HIV acquisition rates.
HIV criminalization is another manifestation of a broader agenda which has attempted to control the bodies, the sexuality, and the desires of queer and trans people and cisgender women, especially those who are low income and/or from communities of color. This is the same agenda that plays out in attempts to control women's access to abortion and contraception and reproductive decisions. This not only includes denying low income women abortion services through Medicaid but the criminalizing of pregnant women who are drug users. The sexual and reproductive rights of communities of color, LGBTQ folks, and women has been policed and criminalized throughout the history of this country. Policies based on restricting our body autonomy, stirring up homo- and transphobia, and spreading HIV-related fears have never been and will never be just or sound public policy.
Alternatives to Criminalization: Towards Restorative Justice and Healing
We acknowledge the HIV epidemic has caused immense pain to many in our communities. As a society, we must be intentional about supporting and providing healing for those who have been affected by HIV. We firmly believe that HIV criminalization does not serve to meet these ends. Prisons will not save us. Criminalization is never a solution. Instead, we call for awholistic approach based on restorative justice principles. Rather than resorting to criminalizing sexuality of people living with HIV, we should treat HIV as an issue of public health, individual health, and human rights and dignity. We must ensure that everyone who is living with HIV (and those who are not) have access to quality and affordable healthcare. As stated above, data shows that suppressing the viral load of a person living with HIV through effective care and treatment reduces the chances of HIV transmission to zero, even if condoms are not used. If states like Missouri are seriously concerned about reducing HIV transmission, they would do better to focus their resources on ensuring their residents living with HIV have access to high-quality, nonstigmatizing, trauma-informed, affordable healthcare. Instead they perpetuate a political agenda that cuts lives short and violates human rights, especially for people of color and those living in poverty, by refusing to expand Medicaid.
Even more importantly than individual actions, we must push for societal changes to the norms and stereotypes that inhibit sexual autonomy and encourage higher risk behaviors. We must advocate for sex education that challenges dominant paradigms around gender norms and heteronormativity. Thus, comprehensive sex educationrooted in modern medical science, sex positivity, and harm reduction, and inclusive of all sexualities and genders is crucial. We must address systemic discrimination that places people at risk for housing, food and employment insecurity. We must advocate for sex education that challenges dominant paradigms around gender norms and heteronormativity. We must address systemic discrimination that places people at risk for housing, food and employment insecurity. We should demand media accountability on pathologized portrayals of Black, brown, and queer bodies and sexuality.
We should demand media accountability on pathologized portrayals of Black, brown, and queer bodies and sexuality.
Demanding accountability
HIV criminalization laws are intricately tied to histories of racism, sexism, and homophobia. These forces in the present continue to enact injustice and perpetuate these laws. For this reason, we call for greater engagement of LGBT and racial justice organizations and leaders in HIV decriminalization advocacy. We know various local, state, and national organizationsand individuals have already stepped up to the plate, but more boots on the ground are needed to fight back against these laws. LGBT and racial justice organizations must take more leadership around this issue by resourcing advocacy, defense litigation, attempts to repeal these laws at the state level, and drawing attention to HIV criminalization as a practice grounded in homophobia, racism, and sexual and reproductive oppression.
We are heartbroken at what has happened to Michael Johnson, but we are no less determined to fight for him, fight for his freedom, and the freedom of all our brothers and sisters incarcerated under HIV criminalization laws. We are also equally committed to standing in solidarity with all movements committed to ending oppression from the dominant culture of policing and criminalizing vulnerable communities. Together we become more powerful. We must resist. We will resist. We resist.
Black is not a crime.
LGBTQ is not a crime.
HIV is not a crime
To sign your organization on to this statement, click this link
Dave Wilson - Your Campaign of HATE Will Not Win - UPDATE
Once again the haters in Houston have nothing better to do but to attack the LGBT community, more specifically the Transgender community. I for one am tired of all of Anti - LGBT rhetoric that you and your elected and clergy friends are trying to use against me and community. At this point in the game YOU LOST, you lost against HERO (so far), and you damn sure lost your fight against LGBT community gaining the right to get married.
For some time you have made attempts to disrupt the live of millions of people, not just LGBT folks. Your last attempt at office was a sham, yes, you won at the expense of a predominantly Black community who lacks the resources to research and properly vet who they are voting for. You are a predator of the worst kind, with full awareness of the cost associated with your attempts to get were you want in order to do nothing with the power you desire. In Wilson's own words, when challenged by KHOU during their story on his PRETENDING TO BE BLACK to get elected to the HCC Board of Trustees: "Every time a politician talks, he's out there deceiving voters."
"Wilson, a gleeful political troublemaker, printed direct mail pieces strongly implying that he's black. His fliers were decorated with photographs of smiling African-American faces -- which he readily admits he just lifted off websites -- and captioned with the words Please vote for our friend and neighbor Dave Wilson.
One of his mailers said he was Endorsed by Ron Wilson, which longtime Houston voters might easily interpret as a statement of support from a former state representative of the same name who's also African-American. Fine print beneath the headline says Ron Wilson and Dave Wilson are cousins, a reference to one of Wilson's relatives living in Iowa." - KHOU
Don't you mean that every time you "talk" that you are "out there deceiving voters?' It is no secret that you are strongly anti LGBT, you thought that you were hiding in plain sight. We know who you are sir and how you have tried to spread hate for years. Parading around as a Democrat and a Republican with the same consistent platform of half assed, disproven, hate-filled anti LGBT rhetoric. It may have been successful with that 2001 proposition to ban city employee benefits for same-sex or unmarried partners, but mark my words, you WILL GO DOWN AND HARD.
From the Houston Chronicle in Wilson's own words: (11/7/2001)
"Gay rights, a theme in several city elections of the last 25 years, was the burning issue. Calling homosexuality a sin, sign company owner and conservative activist Dave Wilson forced the city charter change proposal onto the ballot by gathering signatures from more than 20,000 voters.
He started the movement in February after Mayor Lee Brown, a gay-rights supporter, proposed that the same-sex partners of city workers be eligible for the benefits. Brown put the idea on hold later with the agreement of gay leaders. But Wilson, a former candidate for mayor and councilman, pressed on with the referendum.
"We are very pleased at the outcome," Wilson said Tuesday night. "There are some things that need to happen to continue with the effort for family values and marriage in this community."
Wilson, outspent by his campaign opposition, said he would have piled up a more impressive victory if "the Christian community" had joined his cause publicly.
"Questions need to be asked" about the reluctance of conservatives to campaign for his proposition, he said.
Wilson said he spent about $100,000 of his own money on the campaign through his organization called Houstonians for Family Values, and raised about $30,000 more. His opposition, People for a Fair Houston, spent roughly three times more and was steered by Grant Martin, re-election campaign manager for Annise Parker, City Council's only openly gay member.
Martin said Tuesday night that a victory for Wilson meant that advocates of equal rights for gays need to do a better job of educating voters about such issues."
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Sloths live in tropical rain forests. They can be found living in Central and South America. They have long arms and resemble monkeys – but unlike monkeys' sloths are very (very) slow moving! Three-toed sloths are the world's slowest animal. Sloths are more closely related to anteaters and armadillos.
According to National Geographic, scientists think that sloths are slow moving to avoid being noticed by predators. They often will have algae growing on them too – which helps hid them.
The Sloth Sanctuary of Costa Rica educates the public about the importance of the sloth to the environment. Baby sloths that get left behind for whatever reason are cared for at the sanctuary.


Pedrali Designer Profile Eugeni Quitllet designer of the new Remind chair.
Pedrali Designer Profile Odo Fioravanti new release Dome stool and Babila Stool available now at Business Interiors.
Pedrali Designer Profile Patrick Norguet. Designer of the new Fox range of chairs.


THE MR. DP STORY - Schmuckatelli Co.
The laughing skull, or Mr. DP (Dangerous Places) as he's known, is something I created to give out to military and rebel check points. I developed Mr. DP as a sort of "Kilroy was here" that people would identify with me if someone came looking.
The logo on T-shirts and hats was soon worn by rebel commanders, special forces teams, private military groups, contractors and travelers to war zones. You will even see Mr. DP in famous bars from Afghanistan to Djibouti to Burma. Mr. DP represents those who go past the limits, across borders and constantly push the boundaries.


text:
Arkansas Highway 309 (AR 309 and Hwy. 309) is a designation for two state highways in Western Arkansas. One route of  runs from Yell County Route 28 (CR 28) at Blue Mountain Lake northeast to Highway 10 at Waveland. A second segment begins at Highway 10 in Havana and winds northwest through the Ozark National Forest to Highway 23 at Webb City via Paris. A portion of the second route is designated as the Mount Magazine Scenic Byway, one of eleven Arkansas Scenic Byways maintained by the Arkansas State Highway and Transportation Department (AHTD).

Route description

Blue Mountain Lake to Waveland
Highway 309 begins at Yell County Route 28 near Blue Mountain Lake in Yell County. The route runs east and turns north to terminate at Highway 10. This alignment does not cross any other state highways. An average daily vehicle count from 2010 indicated that 630 vehicles per day use this segment of Highway 309.

Havana to Webb City

A second, longer routing of Highway 309 begins in Havana at Highway 10. The route begins north through the Ozark National Forest. The route runs near the Cove Creek Bridge, Burnett Springs, and the Cove Lake Bathhouse and Cove Lake Spillway Dam-Bridge near Cove Lake. The highway also passes over the Cove Creek Tributary Bridge prior to exiting the forest northbound to Paris.
Along this route lies the small town of Roseville. According to local folklore Roseville missed out on being the capital of Arkansas by only one vote. Whither this is true or not has never been proven, but this story has been passed down from generation to generation and survives til this day.

Upon entering the northern county seat of Logan County Highway 309 forms a concurrency with Highway 109 in south Paris, continuing north to Highway 22. This junction also contains Logan County Courthouse and the Paris Commercial Historic District on the National Register of Historic Places. Highway 309 continues north past the Methodist Episcopal Church, and Paris Post Office to exit town heading north. The route runs north, briefly along the Arkansas River before it enters Franklin County. The route runs almost due north to Webb City, where it terminates at Highway 23.

Mount Magazine Scenic Byway

A segment of  winds from Havana north through the Ozark National Forest to an area south of Paris. The route passes by Mount Magazine, which at  is Arkansas's highest point. Recreation areas include Cove Lake and Mt. Magazine Recreation Areas, with views of Blue Mountain Lake available from an overlook. Many current camp sites are former home sites that were bought by the government during the depression.

Major intersections
Mile markers reset at concurrencies.

|-
| colspan="4" style="text-align:center;"|  concurrency north,

Former route

Highway 309 (AR 309, Ark. 309, and Hwy. 309) is a former route in Ozark. The route of  began at Highway 23 and ran east to Highway 219. It was redesignated as Highway 96 in 1991 due to traveler confusion with the existing section of Highway 309.

History
Highway 309 was first added to the state highway system as a  access road to the Ozark Municipal Airport by the Arkansas State Highway Commission on March 26, 1975. The Commission requested a study regarding extending the route east to Highway 219 on July 31, 1975, and requested an update to that study on March 22, 1978. Although the Commission requested the study's findings be reported at a later meeting, no record of the report's findings exist in subsequent meeting minutes between 1978 and June 1, 1991, when the Commission authorized the extension to Highway 219. Three months later, the Commission redesignated the highway as Highway 96 due to confusion between the two eastbound routes with junctions on Highway 23 within a few miles of each other.

Major intersections

See also

 List of state highways in Arkansas
 Arkansas Scenic Byways
 Mount Magazine State Park

References

External links

309
Transportation in Franklin County, Arkansas
Transportation in Logan County, Arkansas
Transportation in Yell County, Arkansas
Ozark–St. Francis National Forest

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text:
The Verdict on Pickled Veg…
Home /Food, Health & Wellness/The Verdict on Pickled Veg…
Food,Health & Wellness |
Pickles have been around for thousands of years, dating as far back as 2030 BC when cucumbers from India were pickled in the Tigris Valley and continues to this very day.
The word "pickle" comes from the Dutch pekel or northern German pókel, meaning "salt" or "brine," two very important components in the pickling process. Throughout history pickling was a necessity, as it was the best way to preserve food for a long period of time. As one of the earliest mobile foods, pickles filled the stomachs of hungry sailors and travellers, while also providing families with a source of food during the cold winter months. But with the advent of modern technology, do we still need to pickle? And more importantly, is it good for you?
We have oft been told that "fresh is best" where food is concerned and that preserving food could erode its nutritional value or worst still, be unhealthy to consume. What then of pickled vegetables such as Kimchi?
Photo by Portuguese Gravity on Unsplash
Here are 5 reasons why pickling may still be relevant.
Low Fat Content
Like most vegetables, pickles are mostly water and have very low-fat content as a result. They are therefore useful to augment a low-fat diet
2. High Vitamin Content
Pickled vegetables have a high concentration of vitamins because the salty brine draws water out from the pickles leaving the contents rich in vitamins per serving. Contrary to what may be said, the pickling process does not destroy the vitamin content.
For example, cucumbers are high in an antioxidant called beta-carotene, which your body turns into vitamin A. Carotene is a powerful compound that's been shown to help lower your chances of dying of heart disease, cancer or respiratory diseases, and other conditions.
3. Good for Digestion
Fermented vegetables such as Kimchi can help keep your gut healthy. Fermented pickles are full of good bacteria called probiotics which are important for a healthy gut.
But most pickles on supermarket shelves are not fermented, which uses yeast, bacteria, and other microbes to preserve foods. Instead, pickles often get their sharp tang from soaking in a brine of vinegar and spices. So if you are after maximum gut health, look out for labels that say "naturally fermented." When you open the jar, you should see bubbles on the surface, a sign of live bacteria inside.
Photo by Ksenia Chernaya from Pexels
4. Replacing Electrolytes
Some athletes have said that pickle juice relieves muscle cramps and tiredness after a session of heavy exercise due to its claimed abilities to replace electrolytes which are minerals that are vital for health and survival. Electrolytes spark cell function throughout the body. They support hydration and help the body produce energy. They're also responsible for stimulating muscle contractions, including those that keep your heart beating.
5. Keep Sugar Spikes in Check
Due to the presence of vinegar in many pickled vegetables, it is said that pickled vegetables may help regulate blood sugar levels and reduce the risk of diabetes
With everything, however, there are pitfalls. Pickled vegetables contain almost zero protein and other foods must be consumed alongside to make up for this lack.
It should also be noted that pickled vegetables contain very high levels of salt. For example, just one large dill pickle has more than 2/3 of the ideal amount of sodium an average adult should have for the entire day. Excessive salt in one's diet can raise the risk of high blood pressure which can also increase the chances for heart attacks, strokes, diabetes and kidney disease. Too much sodium can also leach calcium from your bones which in turn can weaken the bones
As always, please consult a qualified nutritionist or doctor if in doubt.
Why You Should Always Wash Your Salad Thoroughly
Monophasic Sleep versus Biphasic Sleep
TAGS: digestion fresh is best gut health kimchi nutrition pickle pickled vegetable pickles preserved food preserved vegetable vegetable
9 changes in food consumption…

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text:
4 Classes in the Humanities for Life Sciences Majors
Greek, Latin, Reasoning and Ethics Reinforce Careers in Research and Medicine
Have you ever played word games with a professional in a life sciences field? If so, you know the sheer frustration of losing to words such as "fasciculation" or "crepitus."
5 College Mashups That Make Great Majors
Broaden Your Career Appeal by Expanding Across Traditional Boundaries
You know the pop culture phenomenon that mashes up existing video, song and more to create something new and exciting? UC Davis does that — we think and work across traditional boundaries. In fact, we're a leader in the mashups that, in the academic world, are called interdisciplinary studies. If you want to innovate, help contribute to solving the world's most pressing problems and open up your career opportunities, here are five major mashups to check out.
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text:
\section{Introduction}\label{sec:1}
Triangular norms and triangular conorms, in shortly t-norms and s-norms, are appeared in several areas of mathematics. The beginning of them goes back to \cite{menger,sch}. One of the areas that t-norms and s-norms are appeared, is many-valued logics. Indeed, a t-norm (s-norm)
could be seen as a generalization for the interpretation of the conjunction connective (disjunction connective) \cite{hol,als}.

Basic logic which is introduced by H{\'a}jek in the early of 1998 \cite{hajek98} is known as the logic of continuous t-norms. The algebraic counter part
of the propositional basic logic is BL-algebras. Indeed, the Lindenbaum algebra of equivalent formulas of
propositional basic logic is a BL-algebra. MV-algebras, which introduced by Chang \cite{chang1959new} to prove the completeness theorem for \lo logic, are special kind of BL-algebras. A more general algebraic structure originated in logics without contractions is residuated lattice. The oldest such structure which is appeared in classical logic is Boolean algebra.

This algebraic structures are studied from algebraic and topological point of view. Algebra studies the property of operations and algorithmic computations of a space, while topology provide a framework to understanding the geometric properties of a space.
After introducing the concept of BL-algebras \cite{hajek98}, algebraic and topological properties of them, is one of the interesting research areas.

Bozooei et.al in \cite{zah-bor-2016, bor-rez-kou-2011} introduce the notion of topological BL-algebras and then in \cite{bor-rez-kou-2012} they study the metrizability of BL-algebras.
The objective of this article is introducing a natural topology on BL-algebras which makes them topological algebras. This article is an extended and further development of the author
conference paper \cite{khatami2018}, in which a metric on $[0,1]$ as a dual of BL-algebras studied.

When $[0,1]$ is endowed to be a dual of BL-algebra, ie an algebra
$(L, \dota, \dotv, \star, \dotto , 0, 1)$ of type $(2,2,2,2,0,0)$ which satisfies the
conditions in Definition \ref{dbl-algebra}, then $\star$ present as an s-norm. We show that
when $\star$ is weaker than the \lo s-norm, then the mapping
$d_\star(a,b)=(a\dotto b)\star(b\dotto a)$ becomes a metric on $[0,1]$. Furthermore, $d_\star$ induced a metric $d_{\bigstar}$ on $L^2$ and all of the operations becomes continuous functions from
$(L^2, d_{\bigstar})$ into $(L, d_\star)$. We extend the induced topology of $d_\star$
to a topology on arbitrary BL-algebra $\mathcal{L}$ and it's shown that $\mathcal{L}$ becomes a
topological BL-algebra.

The rest of the paper organized as follows. Section 2 presents a summary about t-norms, s-norms and BL-algebras. Section 3 introduces the concept of dual of BL-algebras and describes a metric
on $[0,1]$ as a dual of BL-algebra and section 4 proves that it is a topological algebra. Section 5
explains a topology on dual of BL-algebras which
makes them topological algebras, and finally a natural topology on BL-algebras
which makes them topological algebras
presented in section 6.
\section{\bf preliminaries}\label{sec:per}
\subsection{t-norms and s-norms}\hfill\\
Recall that a triangular norm, in shortly a t-norm, is a binary function $T$ from $[0,1]^2$ into $[0,1]$ which is associative, commutative, non-decreasing on both arguments and $T(1,x)=x$ for all $x\in[0,1]$.
If the boundary condition of a t-norm is reversed, it is called a t-conorm or an s-norm. Thus an s-norm is an associative, commutative, non-decreasing function $S$ from the unite square into the unite interval satisfying for all $x\in[0,1]$ the boundary condition $S(0,x)=x$.

A t-norm $T$ and an s-norm $S$ are called dual if $S(x,y)=1-T(1-x,1-y)$ for all $x,y\in[0,1]$. Bellow,
the most important t-norms and their dual s-norms are listed in Table \ref{tnor}.
\begin{table}[ht!]
\centering
\caption{
\lo, \g, product, and drastic t-norm and s-norm
}
\label{tnor}
\begin{tabular}{l l l}
\toprule
\rowcolor{black!20}
t-norm&s-norm
\\
\midrule
$T_L(x,y)=\max\{0,x+y-1\}$&$S_L(x,y)=\min\{1,x+y\}$
\\
\midrule
$T_G(x,y)=\min\{x,y\}$&$S_G(x,y)=\max\{x,y\}$
\\
\midrule
$T_\pi(x,y)=x.y$&$S_\pi(x,y)=x+y-x.y$
\\
\midrule
$T_d(x,y)=\left\{\begin{array}{ll}
\min\{x,y\}&\max\{x,y\}=1\\
0&\mbox{otherwise}
\end{array}\right.$
&
$S_d(x,y)=\left\{\begin{array}{ll}
\max\{x,y\}&\min\{x,y\}=0\\
0&\mbox{otherwise}
\end{array}\right.$
\\
\midrule
\end{tabular}
\end{table}

A t-norm $T_1$ is called weaker than $T_2$, in symbols $T_1\le T_2$, whenever $T_1(x,y)\le T_2(x,y)$ for all $x,y\in[0,1]$. A similar notion used for s-norms. Obviously for t-norms and s-norms in Table \ref{tnor},
\begin{center}
$T_d\le T_L\le T_\pi\le T_G$ and $S_G\le S_\pi\le S_L\le S_d$.
\end{center}
Also for any t-norm $T$ and s-norm $S$, one could easily verify that
\begin{center}
$T_d\le T\le T_G$ and $S_G\le S\le S_d$.
\end{center}
\subsection{BL-algebras}\hfill\\
In many-valued logics, t-norms and s-norms sometimes play the role of the interpretation of
the conjunction and disjunction connective. In 1998, H{\'a}jek introduced a many-valued logic, namely basic logic, which is based only on continuous t-norms \cite{hajek98}. If $Prop$ is generated from a set of atomic propositions $P$ by formal operations $\{\&, \to, \bot\}$ and $e_0: P\to[0,1]$ be a function, then there is a unique extension $e$ of $e_0$, called an evaluation, satisfying the following rules:
\begin{center}
$e(\bot)=0$, $e(\varphi \&\psi)=T\left(e(\varphi),e(\psi)\right)$, and $e(\varphi\to \psi)=R_T(e(\varphi),e(\psi))$,
\end{center}
where $R_T$ is defined as follows,
\begin{center}
$z\le R_T(x,y)$ iff $T(z,x)\le y$ for all $z,y,z\in [0,1]$.
\end{center}
The algebraic counter part of a theory in basic logic, forms an algebra, called BL-algebra.
A BL-algebra is an algebra $\mathcal{L}=(L, \wedge, \vee, *, \too , 0, 1)$ of type $(2,2,2,2,0,0)$ satisfying the following properties:
\begin{enumerate}[leftmargin=9mm,label={BL\arabic*}]
  \item $(L, \wedge, \vee, 0 ,1)$ is a bounded lattice with greatest element $1$ and smallest element $0$,\label{bl1}
  \item $(L, *, 1)$ is an Abelian monoid,\label{bl2}
  \item $\too$ is the residua of $*$, i.e., $c\le a\too b$ iff $c*a\le b$ for all $a,b,c\in L$,\label{bl3}
  \item $a\wedge b=a*(a\too b)$ for all $a,b\in L$,\label{bl4}
  \item $(a\too b)\vee(b\too a)=1$ for all $a,b\in L$.\label{bl5}
\end{enumerate}
For a theory $\Sigma\subseteq Prop$, let $[\varphi]=\{\psi: T\vdash\varphi\leftrightarrow\psi\}$,
$Lind(\Sigma)=\{[\varphi]: \varphi\in Prop\}$, $\le$ is defined by
$[\varphi]\le[\psi]$ iff $\Sigma\vdash(\varphi\to\psi)$, and $[\top]=[\bot\to\bot]$.
Then $(Lind(\Sigma),\le,[\bot],[\top])$ forms a bounded lattice that by the following operations becomes a BL-algebra.
\begin{center}
$[\varphi]*[\psi]=[\varphi\&\psi]~~~~,~~~~[\varphi]\rightarrowtail[\psi]=[\varphi\to\psi]$.
\end{center}
\begin{proposition}\label{blprop}
The following properties are hold in every BL-algebra $\mathcal{L}$.
\begin{enumerate}[leftmargin=11mm,label={B\arabic*}]
\item\label{b1} $a*b=b*a$ and $(a*b)*c=a*(b*c)$,\label{f1}
\item\label{b2} $a*0=0$,\label{f2}
\item\label{b3} $a*(a\too b)\le b$ and $a\le (b\too(a*b)$,\label{f3}
\item\label{b4} $a\le b$ iff $a\too b=1$,\label{f4}
\item\label{b5} if $a\le b$ then $a*c\le b*c$, $c\too a\le c\too b$, and $a\too c\ge b\too c$,\label{f5}
\item\label{b6} $(a\vee b)*c=(a*c)\vee(b*c)$,\label{f6}
\item\label{b7} $a*b\le a$ and $a\le b\too a$,\label{f7}
\item\label{b8} $a\vee b=\big((a\too b)\too b\big)\wedge\big((b\too a)\too a\big)$,\label{f8}
\item\label{b9} $(a\too b)\le \big((b\too c)\too(a\too c)\big)$,\label{f9}
\item\label{b10} $(a\too b)*(b\too c)\le(a\too c)$,\label{f10}
\item\label{b11} $a\too(b\too c)=(a*b)\too c$,\label{f11}
\item\label{b12} $a\too(b\too c)=b\too(a\too c)$,\label{f12}
\item\label{b13} $a\too a=1$,\label{f13}
\item\label{b14} $a\too b\le(a*c)\too(b*c)$,\label{f14}
\item\label{b15} $(a\too b)*(c\too d)\le(a*c)\too(b*d)$,\label{f15}
\end{enumerate}
\end{proposition}
\begin{proof}
See \cite[Cahpter 2]{hajek98}
\end{proof}
When the continuous scale $[0,1]$ endowed to be a BL-algebra, the binary operator $*$ becomes a continuous t-norm on $[0,1]$ \cite{godo99}. The BL-algebra on the real segment $[0,1]$ which is defined by  continuous t-norms, is called t-algebra.
\section{\bf dual of BL-algebras, s-algebras, and the natural metric on s-algebras}\label{sec:dual}
Now, we introduce a dual notion for BL-algebras. For lattice join and meet operators we use a dual notion also. Thus in a lattice $(L, \le, \dota, \dotv)$,
$a\le b$ iff $a\dotv b=a$. So, $a\dotv b=\inf\{a,b\}$ and $a\dota b=\sup\{a,b\}$.
\begin{defn}\label{dbl-algebra}
A DBL-algebra, is an algebra
$\mathcal{L}=(L, \dota, \dotv, \star, \dotto , 0, 1)$ of type $(2,2,2,2,0,0)$ satisfies the following conditions:
\begin{enumerate}[leftmargin=13mm,label={DBL\arabic*}]
  \item $(L, \dota, \dotv, 0 ,1)$ is a bounded lattice
  with greatest element $1$ and smallest element $0$,\label{db1}
  \item $(L, \star, 0)$ is an Abelian monoid,\label{db2}
  \item $\dotto$ is the residua of $\star$, i.e., $a\ge b\dotto c$ iff $a \star b\ge c$ for all $a,b,c\in L$,\label{db3}
  \item $a\dota b=a\star(a\dotto b)$ for all $a,b\in L$,\label{db4}
  \item $(a\dotto b)\dotv(b\dotto a)=0$ for all $a,b\in L$.\label{db5}
\end{enumerate}
\end{defn}
\begin{theorem}
Let $\mathcal{L}$ be a DBL-algebra. then the following properties hold:
\begin{enumerate}[leftmargin=11mm,label={D\arabic*}]
\item $a\star b=b\star a$ and $(a\star b)\star c=a\star (b\star c)$,\label{d1}
\item $a\star 1=1$,\label{d2}
\item $a\star (a\dotto b)\ge b$ and $a\ge b\dotto(a\star b)$,\label{d3}
\item $a\ge b$ iff $a\dotto b=0$,\label{d4}
\item if $a\ge b$ then $a\star c\ge b\star c$, $c\dotto a\ge c\dotto b$, and $a\dotto c\le b\dotto c$,\label{d5}
\item $(a\dotv b)\star c=(a\star c)\dotv(b\star c)$,\label{d6}
\item $a\star b\ge a$ and $a\ge b\dotto a$,\label{d7}
\item $a\dotv b=\big((a\dotto b)\dotto b\big)\dota\big((b\dotto a)\dotto a\big)$,\label{d8}
\item $(a\dotto b)\ge \big((b\dotto c)\dotto(a\dotto c)\big)$,\label{d9}
\item $(a\dotto b)\star(b\dotto c)\ge(a\dotto c)$,\label{d10}
\item $a\dotto(b\dotto c)=(a\star b)\dotto c$,\label{d11}
\item $a\dotto(b\dotto c)=b\dotto(a\dotto c)$,\label{d12}
\item $a\dotto a=0$,\label{d13}
\item $a\dotto b\ge(a\star c)\dotto(b\star c)$,\label{d14}
\item $(a\dotto b)\star (c\dotto d)\ge(a\star c)\dotto(b\star d)$,\label{d15}
\end{enumerate}
\end{theorem}
\begin{proof}
The proof is similar to the proof of Proposition \ref{blprop}. Let's remind it.
\begin{enumerate}[leftmargin=11mm,label={D\arabic*})]
\item Follows from \ref{db2}.
\item Since $a\ge 0$, so $a\star 1\ge 0\star 1$. But
\ref{db2} implies that $1\star 0=1$. Thus $a\star 1=1$.
\item Since $a\dotto b\ge a\dotto b$ by \ref{db3}, $(a\dotto b)\star a\ge b$. Again, since
$a\star b\ge a\star b$, by \ref{db3}, $a\ge b\dotto(a\star b)$.
\item $a\ge b$ iff $0\star a\ge b$ which is by adjointness is equivalent $0\ge a\dotto b$, means that
$a\dotto b=0$.
\item Assume that $a\ge b$. We have the followings.
\begin{itemize}
  \item By \ref{d3} $a\ge b\ge c\dotto(b\star c)$ and so adjointness condition (\ref{db3}) implies that $a\star c\ge b\star c$.
  \item By \ref{d3} $c\star (c\dotto a)\ge a\ge b$. Thus $(c\dotto a)\star c\ge b$. Therefore  by adjointness, $c\dotto a\ge c\dotto b$.
  \item By the first inequality of \ref{d5} $a\star (b\dotto c)\ge b\star (b\dotto c)$. On the other hand by \ref{d3} $b\star (b\dotto c)\ge c$. So, $a\star (b\dotto c)\ge c$. Now, using \ref{d1} and \ref{db3} we have $b\dotto c\ge a\dotto c$.
\end{itemize}
\item Since $a\ge a\dotv b$ \ref{d5} implies that $a\star c\ge (a \dotv b)\star c$.
A similar argument show that $b\star c\ge (a \dotv b)\star c$. So,
$(a\star c)\dotv (b\star c)\ge (a \dotv b)\star c$. For the reverse inequality, since
$a\star c\ge (a\star c)\dotv (b\star c)$ by adjointness
$a\ge c\dotto\big((a\star c)\dotv (b\star c)\big)$. Similarly,
$b\ge c\dotto\big((a\star c)\dotv (b\star c)\big)$. So,
$a\dotv b\ge c\dotto\big((a\star c)\dotv (b\star c)\big)$ that by adjointness implies that
$(a\dotv b)\star c\ge (a\star c)\dotv (b\star c)$.
\item Since $b\ge 0$, applying \ref{d5} and \ref{db2} we have $a\star b\ge a\star 0=a$.
Furthermore, by adjointness this means that $a\ge b\dotto a$.
\item Let $A=\big((a\dotto b)\dotto b\big)\dota\big((b\dotto a)\dotto a\big)$. Applying
\ref{db2}, \ref{db4}, \ref{d6}, and the facts that $A\ge a\dotto(a\dotto b)$,
$A\ge b\dotto(b\dotto a)$, and finally using \ref{d3} we have
\begin{eqnarray*}
 A&=&A\star 0=A\star \big((a\dotto b)\dotv(b\dotto a)\big)
  =\big(A\star(a\dotto b)\big)\dotv\big(A\star(b\dotto a)\big)\\
  &\ge&\big[\big((a\dotto b)\dotto b\big)\star(a\dotto b)\big]\dotv\big[\big((b\dotto a)\dotto a\big)\star(b\dotto a)\big]
  \ge b\dotv a.
\end{eqnarray*}
Conversely,
$(a\dotto b)\star(a \dotv b)=\big((a\dotto b)\star a\big)\dotv\big((a\dotto b)\star b\big)
\ge b\dotv b=b$ by \ref{d6}, \ref{d3}, and \ref{d7}. Now, by adjointness
$(a \dotv b)\ge (a \dotto b)\dotto b$. A similar argument show that
$(a \dotv b)\ge (b \dotto a)\dotto a$. So,
$(a \dotv b)\ge \big((a \dotto b)\dotto b\big)\dota\big((b \dotto a)\dotto a\big)=A$.
\item Using \ref{d3} two times we have
$a \star(a\dotto b)\star (b\dotto c)\ge b\star(b\dotto c)\ge c$, which means that
$(a\dotto b)\star (b\dotto c)\star a\ge c$.
Now, using adjointness two times have the result.
\item Apply adjointness on \ref{d9}.
\item For any point $x$ of $L$ we have
$x\ge a\dotto(b\dotto c)$ iff $x\star a\ge b\dotto c$ iff $(x\star a)\star b\ge c$ iff
$x\star (a\star b)\ge c$ iff $x\ge (a\star b)\dotto c$. Since $x$ is arbitrary we obtain the result.
\item Using \ref{d11} twice we get
$a\dotto(b\dotto c)=(a\star b)\dotto c=(b\star a)\dotto c=b\dotto(a\dotto c)$.
\item By \ref{d7} $a\ge 0\dotto a$. Hence \ref{d4} implies that $a\dotto(0\dotto a)=0$. Applying \ref{d12} we get $0\dotto(a\dotto a)=a\dotto(0\dotto a)=0$. Again using \ref{d4}
    we deduced that $a\dotto a=0$.
\item By \ref{d3} $\big(a\star(a\dotto b)\big)\star c\ge b\star c$ which means that
$(a\dotto b)\star(a\star c)\ge b\star c$. Now, adjointness give
$a\dotto b\ge(a\star c)\dotto(b\star c)$.
\item By \ref{d3},
$\big(a\star (a\dotto b)\big)\star\big(c\star(c\dotto d)\big)\ge b\star d$. So,
$\big((a\dotto b)\star(c\dotto d)\big)\star(a\star c)\ge b\star d$ which by adjointness leads to
$(a\dotto b)\star(c\dotto d)\ge(a\star c)\dotto(b\star d)$.
\end{enumerate}
\end{proof}
Assume that $S$ be a continuous s-norm. The residua of $S$ is the unique operator $R_S:[0,1]^2 \to [0,1]$, defined by the adjoint property,
\begin{center}
for all $a, b, c\in[0,1]$, $S(a,b)\ge z$ iff $a\ge R_S(b,c)$.
\end{center}
Continuity of $S$ implies that $R_S(a,b)=\min\{c: S(c,a)\ge b\}$.
An easy argument show that for any continuous s-norm $S$ and it's residua $R_S$,
$([0,1], \max, \min, S, R_S, 0, 1)$ forms a DBL-algebra, called s-algebra.
\begin{theorem}
If $([0,1], \dota, \dotv, \star, \dotto, 0, 1)$ endowed to be a DBL-algebra, then
$\star$ becomes a continuous t-norm, $\dotto$ would be the residua of $\star$ and
therefore $\dota$ and $\dotv$ becomes the maximum and minimum functions, respectively.
\end{theorem}
\begin{proof}
The proof is similar to the one for BL-algebras. See \cite[Proposition 3]{godo99}.
\end{proof}
For any s-algebra $\mathcal{L}=([0,1], \max, \min, \star, \dotto, 0, 1)$, we introduce a metric on
$[0,1]$ which makes it a topological algebra. Define $d_\star:[0,1]^2\to[0,1]$ by
$d_\star(a,b)=(a\dotto b)\star(b\dotto a)$. The following theorem show that
$d_\star$ is a metric on $[0,1]$.
\begin{theorem}\label{metric}
Let $\star$ be a continuous s-norm and $\dotto$ be the residue of $\star$. Define $d_\star$ by
$d_\star(a,b)=(a\dotto b)\star(b\dotto a)$. Then,
\begin{enumerate}
\item (identity of indiscernibles) $\forall a, b$, $d_\star(a,b)=0$ iff $a=b$,
\item (symmetry) $\forall a, b$, $d_\star(a,b)=d_\star(b,a)$,
\item (transitivity) $\forall a, b, c$, $d_\star(a,b)\le d_\star(a,c)\star d_\star(c,b)$.\label{3}
\end{enumerate}
Furthermore, if $\star$ is weaker than the \lo s-norm, then
$d_\star$ would be a metric on $[0,1]$.
\end{theorem}
\begin{proof}
Clearly, by \ref{d13} $d_\star(a,a)=0$ for $a\in[0,1]$. Furthermore, if $d_\star(a,b)=0$, then
$(a\dotto b)\star(b\dotto a)=0$. Applying \ref{d7} we get $a\dotto b=0$ and $b\dotto a=0$. Now,
\ref{d4} implies that $a\ge b$ and $b\ge a$ that is $a=b$.
Symmetric property of $d_\star$ is clear by \ref{d1}. To prove
\ref{3}, by \ref{d10} $(a\dotto c)\star(c\dotto b)\ge a\dotto b$ and
$(b\dotto c)\star(c\dotto a)\ge b\dotto a$ for arbitrary $a,b,c,\in [0,1]$. Hence,
\begin{eqnarray*}
  d_\star(a,b)&=&(a\dotto b)\star(b\dotto a)\\
  &\le&\big((a\dotto c)\star(c\dotto b)\big)\star\big((b\dotto c)\star(c\dotto a)\big)\\
  &=&\big((a\dotto c)\star(c\dotto a)\big)\star\big((b\dotto c)\star(c\dotto b)\big)\\
  &=&d_\star(a,c)\star d_\star(c,b)
\end{eqnarray*}
Furthermore if $\star$ is weaker than $S_L$, then
\begin{center}
$d_\star (a,b)\le d_\star(a,c)\star d_\star(c,b)
\le S_L\big(d_\star(a,c),d_\star(c,b)\big)\le d_\star(a,c)+d_\star(c,b)$.
\end{center}
\end{proof}
Below, the induced metric arising from tree famous continuous s-norms are listed in Table \ref{snor}.
\begin{table}[ht!]
\centering
\caption{
\lo, \g, and product induced metrics
}
\label{snor}
\begin{tabular}{l l l}
\toprule
\rowcolor{black!20}
s-norm&residua&metric
\\
\midrule
$
\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
S_L(x,y)=\min\{1,x+y\}$&
$
\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
R_L(x,y)=
\left\{\begin{array}{cc}
0& x\ge y\\
y-x& x<y
\end{array}\right.$&
$d_L(x,y)=|x-y|$
\\
\midrule
$
\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
S_G(x,y)=\max\{x,y\}$&
$
\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
R_G(x,y)=
\left\{\begin{array}{cc}
0& x\ge y\\
y& x<y
\end{array}\right.$&
$
\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
d_G(x,y)=
\left\{\begin{array}{cc}
0& x=y\\
\max\{x,y\}& x\ne y
\end{array}\right.$
\\
\midrule
$
\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
S_\pi(x,y)=x+y-x.y$&
$
\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
R_\pi(x,y)=
\left\{\begin{array}{cc}
0& x\ge y\\
\displaystyle\frac{y-x}{1-x}&x<y
\end{array}\right.$
&
$
\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
d_\pi(x,y)=\left\{\begin{array}{cc}
0&x=y\\
\displaystyle\frac{|x-y|}{1-\min\{x,y\}}&x\ne y
\end{array}\right.$
\\
\midrule
\end{tabular}
\end{table}
\section{\bf $[0,1]$ as a topological s-algebra}
The operations of the \lo s-algebra $[0,1]$ are continuous with respect to the Euclidean topology $d_L(x,y)=|x-y|$. However, in general continuity of operations does not hold. Specially,
$R_G$ and $R_\pi$ are not continuous.

In this section, we show that for any continuous s-norm $\star$ and it's residua $\dotto$, all of the operations of the corresponding s-algebra $([0,1],\max, \min, \star, \dotto, 0, 1)$ are continuous with respect to induced topology of the metric $d_\star$ introduced in Theorem \ref{metric}.

The Euclidean topology on $[0,1]^2$ is usually defined by
\begin{center}
$d\big((x_1,x_2),(y_1,y_2)\big)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$,
\end{center}
which is equivalent to the maximum metric $\max\{|x_1-y_1|,|x_2-y_2|\}$ and the taxicab metric
$|x_1-y_1|+|x_2-y_2|$. Here we use a metric, similar to the taxicab metric on $[0,1]^2$ which is made by $\star$ and the metric $d_\star$.
\begin{theorem}\label{metric1}
Let $\star$ , $\dotto$, and $d_\star$ be as in Theorem \ref{metric}. Define
$\mathbf{d}_\star:[0,1]^2\times[0,1]^2\to[0,1]$ by
$\mathbf{d}_\star(\mathbf{a},\mathbf{b})=d_\star(a_1,b_1)\star d_\star(a_2,b_2)$ where
$\mathbf{a}=(a_1,a_2)$ and $\mathbf{b}=(b_1,b_2)$. Then
\begin{enumerate}
  \item $\forall \mathbf{a}, \mathbf{b}\in[0,1]^2$, $\mathbf{d}_\star(\mathbf{a},\mathbf{b})=0$ iff
  $\mathbf{a}=\mathbf{b}$,
  \item $\forall \mathbf{a}, \mathbf{b}\in[0,1]^2$, $\mathbf{d}_\star(\mathbf{a},\mathbf{b})=
  \mathbf{d}_\star(\mathbf{b},\mathbf{a})$,
\item\label{a3} $\forall \mathbf{a}, \mathbf{b}, \mathbf{c}\in[0,1]^2$, $\mathbf{d}_\star(\mathbf{a},\mathbf{b})\le
\mathbf{d}_\star(\mathbf{a},\mathbf{c})\star\mathbf{d}_\star(\mathbf{c},\mathbf{b})$.
\end{enumerate}
Furthermore, if $\star\le S_L$, then $\mathbf{d}_\star$ define a metric on $[0,1]^2$.
\end{theorem}
\begin{proof}
We only check \ref{a3}. Using transitivity of $d_\star$ (Theorem \ref{metric}) and \ref{d1} we have
\begin{eqnarray*}
  \mathbf{d_\star}\big(\mathbf{a},\mathbf{b})&=& d_\star(a_1,b_1)\star d_\star(a_2,b_2)\\
  &\le&\big(d_\star(a_1,c_1)\star d_\star(c_1,b_1)\big)\star\big(d_\star(a_2,c_2)\star d_\star(c_2,b_2)\big)\\
  &=&\big(d_\star(a_1,c_1)\star d_\star(a_2,c_2)\big)\star\big((d_\star(c_1,b_1)\star d_\star(c_2,b_2)\big)\\
  &=&\mathbf{d_\star}(\mathbf{a},\mathbf{c})\star\mathbf{d_\star}(\mathbf{c},\mathbf{b}).
\end{eqnarray*}
Finally, an easy argument like as the proof of Theorem \ref{metric} show that whenever $\star\le S_L$,
$\mathbf{d}_\star$ is a metric on $[0,1]^2$.
\end{proof}
Now, we show that for any continuous s-norm $\star$ and it's residua $\dotto$, the s-algebra $\mathcal{L}=([0,1],\max, \min, \star, \dotto, 0, 1)$ is a topological algebra. To this end we should introduce topologies on $[0,1]$ and $[0,1]^2$ such that all of the operations of $\mathcal{L}$ are continuous functions with respect to these topologies. Note that by \ref{db4} and \ref{d8}
it is enough to show that $\star$ and $\dotto$ are continuous functions. In the case that
$\star$ is weaker than the \lo s-norm, the induced topology of metric $d_\star$ and $\mathbf{d}_\star$ do
the goal.
\begin{theorem}\label{con}
Let $\star$, $\dotto$, $d_\star$, and $\mathbf{d}_\star$ be as in Theorem \ref{metric1}. If
$\star$ is weaker than $S_L$, then $\star:([0,1]^2,\mathbf{d}_\star)\to([0,1],d_\star)$ and $\dotto:([0,1]^2,\mathbf{d}_\star)\to([0,1],d_\star)$ are continuous functions.
\end{theorem}
\begin{proof}
Using \ref{d15} we have
\begin{eqnarray*}
  (a_1\star a_2)\dotto (b_1\star b_2)&\le& (a_1\dotto b_1)\star(a_2\dotto b_2)\\
  &\text{and}&\\
  (b_1\star b_2)\dotto (a_1\star a_2)&\le& (b_1\dotto a_1)\star(b_2\dotto a_2).
\end{eqnarray*}
Now, applying \ref{d5} twice we get
\begin{center}
$\medmuskip=0mu
\thinmuskip=0mu
\thickmuskip=0mu
\big((a_1\star a_2)\dotto (b_1\star b_2)\big)\star
\big((b_1\star b_2)\dotto (a_1\star a_2)\big)
\le\big((a_1\dotto b_1)\star(a_2\dotto b_2)\big)
\star\big((b_1\dotto a_1)\star(b_2\dotto a_2)\big)$,
\end{center}
that is
\begin{equation}\label{a1}
d_\star(a_1\star a_2, b_1\star b_2)\le \mathbf{d}_\star(\mathbf{a},\mathbf{b})
\end{equation}
which means that $\star$ is a continuous function.

For continuity of $\dotto$, by \ref{d9}
$a_1\dotto b_1\ge (b_1\dotto b_2)\dotto (a_1\dotto b_2)$, and therefore by adjointness
\begin{equation}\label{z1}
(a_1\dotto b_1)\star(b_1\dotto b_2)\ge a_1\dotto b_2.
\end{equation}
Again by \ref{d9} $a_1\dotto b_2\ge (b_2\dotto a_2)\dotto (a_1\dotto a_2)$ which beside the inequality \ref{z1} leads to
\begin{center}
$(a_1\dotto b_1)\star(b_1\dotto b_2)\ge (b_2\dotto a_2)\dotto (a_1\dotto a_2).$
\end{center}
Now, applying adjointness we get
\begin{center}
$\big((a_1\dotto b_1)\star(b_1\dotto b_2)\big)\star(b_2\dotto a_2)\ge(a_1\dotto a_2)$,
\end{center}
which beside \ref{d1} implies that
\begin{center}
$\big((a_1\dotto b_1)\star(b_2\dotto a_2)\big)\star(b_1\dotto b_2)\ge(a_1\dotto a_2)$.
\end{center}
Finally, again by adjointness
\begin{equation}\label{z2}
(b_1\dotto b_2)\dotto(a_1\dotto a_2) \le(a_1\dotto b_1)\star(b_2\dotto a_2).
\end{equation}
A similar argument show that
\begin{equation}\label{z3}
(a_1\dotto a_2)\dotto(b_1\dotto b_2) \le(b_1\dotto a_1)\star(a_2\dotto b_2).
\end{equation}
Now using \ref{z2} and \ref{z3} and applying \ref{d5} we have
\begin{eqnarray*}
d_\star (a_1\dotto a_2, b_1\dotto b_2)&=&
\big((b_1\dotto b_2)\dotto(a_1\dotto a_2)\big)\star
\big((a_1\dotto a_2)\dotto(b_1\dotto b_2)\big)\\
&\le&
\big((a_1\dotto b_1)\star(b_2\dotto a_2)\big)\star
\big((a_1\dotto a_2)\dotto(b_1\dotto b_2)\big)\\
&\le&\big((a_1\dotto b_1)\star(b_2\dotto a_2)\big)\star
\big((b_1\dotto a_1)\star(a_2\dotto b_2)\big)\\
&=&\big((a_1\dotto b_1)\star(b_1\dotto a_1)\big)\star
\big((b_2\dotto a_2)\star(a_2\dotto b_2)\big)\\
&=&d_\star(a_1,b_1)\star d_\star(a_2,b_2)\\
&=&\mathbf{d_\star}(\mathbf{a},\mathbf{b})
\end{eqnarray*}
Hence $\dotto$ is a continuous function.
\end{proof}
Now, we want to generalize Theorem \ref{con} for the case that $\star$ is an arbitrary
continuous s-norm.
\begin{defn}\label{top}
Assume that $\star$, $\dotto$, $d_\star$, and $\mathbf{d}_\star$ be as in Theorem \ref{metric1}.
($\star$ is not necessarily weaker than the \lo s-norm).
For $a\in [0,1]$ and $r\in(0,1]$ the $\star$-ball around $a$ of radius $r$ is the set
\begin{center}
$N_r(a)=\{b\in[0,1]: d_\star(a,b)<r\}.$
\end{center}
Similarly the $\bigstar$-ball around $\mathbf{a}\in [0,1]^2$ of radius $r\in(0,1]$ is the set
\begin{center}
$\mathbf{N}_r(\mathbf{a})=\{\mathbf{b}\in[0,1]^2: \mathbf{d}_\star(\mathbf{a},\mathbf{b})<r\}.$
\end{center}
A subset $G$ of $[0,1]$ is called an $\star$-open set if for every $a\in G$ there exists a radius $r>0$ such that $N_r(a)\subseteq G$. $\bigstar$-open subsets of $[0,1]^2$ defined similarly.
\end{defn}
\begin{theorem}\label{topology}
With the notions in Definition \ref{top}, the family of all $\star$-open sets, denoted by $T_\star$ form a topology on $[0,1]$. Similarly, the family of all $\bigstar$-open sets of $[0,1]^2$, denoted by $T_\bigstar$ form a topology on $[0,1]^2$.
\end{theorem}
\begin{proof}
Obviously $\emptyset, [0,1]\in T_\star$.

Assume that $A, B\in T_\star$. If $A\cap B$ is empty, then it is in $T_\star$. Suppose that $A\cap B\ne\emptyset$ and $a\in A\cap B$. Since $A$ and $B$ are $\star$-open sets, there exist
$r_A>0$ and $r_B>0$ such that $N_{r_A}(a)\subseteq A$ and $N_{r_B}(a)\subseteq B$. let
$r=\min\{r_A, r_B\}$. Now, $N_r(a)\subseteq N_{r_A}(a)\cap N_{r_B}(a)\subseteq A\cap B$. So, $A\cap B$ is an $\star$-open set.

Finally, let $\{G_i\}_{i\in I}$ be a family of $\star$-open sets and $G=\cup_{i\in I}G_i$.
If $G$ is empty there is noting to prove. Assume that $G\ne\emptyset$ and $a\in G$. So, there is
$i\in I$ such that $a\in G_i$. Since $G_i$ is an $\star$-open set, there exists $r>0$ such that
$N_r(a)\subseteq G_i\subseteq G$. Thus $G$ is an $\star$-open set.

The second part is also proved by a similar argument.
\end{proof}
Note that when $\star\le S_L$, then topology $T_\star$ ($T_\bigstar$) is the topology
induced by metric $d_\star$ ($\mathbf{d}_\star$).
\begin{theorem}\label{contin}
Let $\star$, $\dotto$, $d_\star$, and $\mathbf{d}_\star$ be as in Theorem \ref{metric1} ($\star$ is not necessarily weaker than the \lo s-norm). Then the mappings $\star:([0,1]^2,T_\bigstar)\to([0,1],T_\star)$ and $\dotto:([0,1]^2,T_\bigstar)\to([0,1],T_\star)$ are continuous functions.
\end{theorem}
\begin{proof}
For continuity of $\star$, consider an $\star$-open set $A\in T_\star$. We must show that the inverse image of $A$, $\star^{-1}(A)$, is an $\bigstar$-open set. For this consider a
point $\mathbf{a}\in \star^{-1}(A)$. So, $\star(\mathbf{a})\in A$, that is
$a_1\star a_2\in A$. But $A$ is an $\star$-open set and hence there exists $r>0$ such that
$N_r(a_1\star a_2)\subseteq A$. We claim that
$\mathbf{N}_r(\mathbf{a})\subseteq \star^{-1}(A)$.

Consider an element $\mathbf{b}\in\mathbf{N}_r(\mathbf{a})$. So, $\mathbf{d}_\star(\mathbf{a},\mathbf{b})<r$. Now, inequality \ref{a1} in the proof of Theorem \ref{con}
implies that
\begin{center}
$d_\star(a_1\star a_2, b_1\star b_2)\le \mathbf{d}_\star(\mathbf{a},\mathbf{b})<r$.
\end{center}
So, $b_1\star b_2\in N_r(a_1\star a_2)\subseteq A$, that is $\star(\mathbf{b})\in A$ or
$\mathbf{b}\in \star^{-1}(A)$. Hence, $\mathbf{N}_r(\mathbf{a})\subseteq \star^{-1}(A)$ which means that
$\star^{-1}(A)$ is an $\bigstar$-open set. Thus, $\star$
is a continuous function.

A similar argument, that use the final argument in the proof of Theorem \ref{con} show that $\dotto$
is a continuous function.
\end{proof}
\section{\bf natural topology on DBL-algebras}\label{DBL}
In this section, we introduce a topology on an arbitrary DBL-algebra which makes it a topological algebras. We need a definition for positive element of DBL-algebras.
\begin{defn}
Let $\mathcal{L}=(L, \dota, \dotv, \star, \dotto , 0, 1)$ be a DBL-algebra.
An element $a\in L$ is called a positive element of $L$, denoted by $a\gg 0$, whenever
for any $b\in L$, $a\dotv b=0$ implies that $b=0$.
\end{defn}
\begin{lemma}\label{ggp}
Let $\mathcal{L}=(L, \dota, \dotv, \star, \dotto , 0, 1)$ be a DBL-algebra.
for any $a,b\in L$,
\begin{enumerate}[leftmargin=11mm,label={G\arabic*}]
  \item $1\gg 0$,
  \item if $a\gg 0$, then $a>0$,
  \item if $b>a\gg 0$, then $b\gg 0$,
  \item\label{g4} if $a\gg 0$ and $b\gg 0$, then $a\dotv b\gg 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[leftmargin=11mm,label={G\arabic*})]
\item For any $x\in L$, since $1\dotv x=x$, so $1\dotv x=0$ implies that $x=0$, that is $1\gg 0$.
\item If by contrary, $a=0$, then for $x\ne 0$, $a\dotv x=0$, which is in contradiction with $a\gg 0$.
\item For an arbitrary $x\in L$, assume that $b\dotv x=0$. Since $b>a$, $b\dotv x\ge a\dotv x$. So,
$a\dotv x=0$ and therefore $x=0$. Thus $b\gg 0$.
\item For an arbitrary $x\in L$, assume that $(a\dotv b)\dotv x=0$. Thus,
$a\dotv(b\dotv x)=0$. Therefore, $b\dotv x=0$. Hence, $x=0$ that is $a\dotv b\gg0$.
\end{enumerate}
\end{proof}
Now, as like as the Definition \ref{top}, we introduce a topology on DBL-algebras.
\begin{defn}\label{top1}
Assume that $\mathcal{L}=(L, \dota, \dotv, \star, \dotto , 0, 1)$ be a DBL-algebra.
Suppose that the mappings $d_\star:L\times L\to L$ and $\mathbf{d}_\star:L^2\times L^2\to L$ defined by
\begin{center}
$d_\star(a,b)=(a\dotto b)\star(b\dotto a)$ and
$\mathbf{d}_\star(\mathbf{a},\mathbf{b})=d_\star(a_1,b_1)\star d_\star(a_2,b_2)$,
\end{center}
respectively, where $\mathbf{a}=(a_1,a_2)$ and $\mathbf{b}=(b_1,b_2)$. For any
$a\in L$ and any positive element $r\in L$, the $\star$-ball around $a$ of radius $r$ is the set
\begin{center}
$N_r(a)=\{b\in L: d_\star(a,b)<r\}.$
\end{center}
Similarly the $\bigstar$-ball around $\mathbf{a}\in L^2$ of radius $r\gg 0$ is the set
\begin{center}
$\mathbf{N}_r(\mathbf{a})=\{\mathbf{b}\in L^2: \mathbf{d}_\star(\mathbf{a},\mathbf{b})<r\}.$
\end{center}
A subset $G$ of $L$ is called an $\star$-open set if for every $a\in G$ there exists a radius $r\gg 0$ such that $N_r(a)\subseteq G$. $\bigstar$-open subsets of $L^2$ defined similarly.
\end{defn}
\begin{theorem}
with the notions in Definition \ref{top1},
\begin{enumerate}
\item $\forall a, b$, $d_\star(a,b)=0$ iff $a=b$,
\item $\forall a, b$, $d_\star(a,b)=d_\star(b,a)$,
\item $\forall a, b, c$, $d_\star(a,b)\le d_\star(a,c)\star d_\star(c,b)$.\label{3}
\end{enumerate}
Furthermore, a similar argument holds for $d_\bigstar$.
\end{theorem}
\begin{proof}
Similar to the proofs of Theorem \ref{metric} and Theorem \ref{metric1}.
\end{proof}
\begin{theorem}\label{topthe}
With the notions in Definition \ref{top1}, the family of all $\star$-open sets
form a topology on $L$ denoted by $T_\star$. Similarly $T_\bigstar=\{A : A~\text{is an}~
\bigstar\text{-open set of}~L^2\}$ is a topology on $L^2$.
\end{theorem}
\begin{proof}
Obviously $\emptyset, L\in T_\star$. Assume that $A, B\in T_\star$. If $A\cap B$ is empty, then it is in $T_\star$. So, let $a\in A\cap B$. Since $A$ and $B$ are $\star$-open sets, there exist
$r_A\gg 0$ and $r_B\gg 0$ such that $N_{r_A}(a)\subseteq A$ and $N_{r_B}(a)\subseteq B$. By \ref{g4},
$r=r_A\dotv r_B\gg 0$. Now, $N_r(a)\subseteq N_{r_A}(a)\cap N_{r_B}(a)\subseteq A\cap B$. Hence, $A\cap B$ is an $\star$-open set. The rest of the proof is similar to the proof of Theorem \ref{topology}.
\end{proof}
The following theorem show that the introduced topologies on $L$ and $L^2$, makes $\mathcal{L}$ a topological algebra.
\begin{theorem}
With the notions in Theorem \ref{topthe},
the mappings $\star:(L^2,T_\bigstar)\to(L,T_\star)$ and $\dotto:(L^2,T_\bigstar)\to(L,T_\star)$ are continuous functions.
\end{theorem}
\begin{proof}
Similar to the proof of Theorem \ref{contin}.
\end{proof}
\section{\bf natural topology on BL-algebras}
In this section, using the duality between BL-algebras and DBL-algebras, we introduce a topology on an arbitrary BL-algebra which makes it a topological algebras. So, this section could be seen as the dual of
Section \ref{DBL}.
\begin{defn}
Let $\mathcal{L}=(L, \wedge, \vee, *, \too , 0, 1)$ be a BL-algebra.
An element $a\in L$ is called strongly less than $1$, denoted by $a\ll 1$, whenever
for any $b\in L$, $a\vee b=1$ implies that $b=1$.
\end{defn}
The following lemma and it's proof, are the dual ones of Lemma \ref{ggp}.
\begin{lemma}
Let $\mathcal{L}=(L, \wedge, \vee, *, \too , 0, 1)$ be a BL-algebra.
for any $a,b\in L$,
\begin{enumerate}[leftmargin=11mm,label={L\arabic*}]
  \item\label{l1} $0\ll 1$,
  \item if $a\ll 1$, then $a<1$,
  \item if $b<a\ll 1$, then $b\ll 1$,
  \item\label{l4} if $a\ll 1$ and $b\ll 1$, then $a\vee b\ll 1$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is the dual ones of the proof of Lemma \ref{ggp}. For example for \ref{l1}, since $0\vee x=x$, so $0\vee x=1$ implies that $x=1$, that is $0\ll 1$.
\end{proof}
For any BL-algebra $\mathcal{L}$, define two operators $\leftrightarrow$ and $\Leftrightarrow$ on $L$ and $L^2$, respectively as follows:
\begin{center}
$a \leftrightarrow b=(a\too b)*(b\too a)~~~~~,~~~~~
\mathbf{a}\Leftrightarrow\mathbf{b}=(a_1\leftrightarrow b_1)*(a_2\leftrightarrow b_2)$.
\end{center}
\begin{remark}\label{r1}
By \ref{b1}, both of $\leftrightarrow$ and $\Leftrightarrow$ are symmetric.
Furthermore, by \ref{b13} $a\leftrightarrow a=1$ and $\mathbf{a}\Leftrightarrow\mathbf{a}=1$
for any $a\in L$ and $\mathbf{a}\in L^2$. In addition an easy consequence of \ref{b10}
implies that for any $a,b,c,\in L$,
$a\leftrightarrow b\ge (a\leftrightarrow c)*(c\leftrightarrow b)$.
Similarly, for any $\mathbf{a}, \mathbf{b}, \mathbf{c}\in L^2$,
$\mathbf{a}\Leftrightarrow\mathbf{b}\ge(\mathbf{a}\Leftrightarrow\mathbf{c})*
(\mathbf{c}\Leftrightarrow\mathbf{b})$. Finally, note that by \ref{b7}
$a\too b\ge a\leftrightarrow b$.
\end{remark}
Now, we introduce a topology on BL-algebras.
\begin{defn}\label{top2}
For
any elements $a,r\in L$ that $r\ll 1$, the $*$-ball around $a$ of radius $r$ is the set
\begin{center}
$B_r(a)=\{b\in L: a\leftrightarrow b>r\}.$
\end{center}
Similarly the $\mathbf{*}$-ball around $\mathbf{a}\in L^2$ of radius $r\ll 1$ is the set
\begin{center}
$\mathbf{B}_r(\mathbf{a})=\{\mathbf{b}\in L^2: \mathbf{a}\Leftrightarrow\mathbf{b}>r\}.$
\end{center}
A subset $G$ of $L$ is called an $*$-open set if for every $a\in G$ there exists a radius $r\ll 1$ such that $B_r(a)\subseteq G$. $\divideontimes$-open subsets of $L^2$ defined similarly.
\end{defn}
\begin{remark}\label{r2}
By Remark \ref{r1}, $a\in B_r(a)$ and similarly $\mathbf{a}\in\mathbf{B}_r(\mathbf{a})$. Moreover,
if $r\ge s$ then $B_r(a)\subseteq B_s(a)$ and
$\mathbf{B}_r(\mathbf{a})\subseteq \mathbf{B}_s(\mathbf{a})$.
\end{remark}
\begin{theorem}
With the notions in Definition \ref{top2}, the family of all $*$-open sets
form a topology on $L$ denoted by $T_*$. Similarly $T_\bigs=\{A : A~\text{is an}~
\bigs\text{-open set of}~L^2\}$ is a topology on $L^2$.
\end{theorem}
\begin{proof}
Obviously $\emptyset, L\in T_*$. Assume that $A, B\in T_*$. if $a\in A\cap B$, then since $A$ and $B$ are $*$-open sets, there exist
$r_A\ll 1$ and $r_B\ll 1$ such that $N_{r_A}(a)\subseteq A$ and $N_{r_B}(a)\subseteq B$. By \ref{l4},  $r=r_A\vee r_B\ll 1$. Since $r\ge r_A$, Remark \ref{r2} implies that $N_r(a)\subseteq N_{r_A}(a)$.
Similarly, $N_r(a)\subseteq N_{r_B}(a)$. Thus $N_r(a)\subseteq N_{r_A}(a)\cap N_{r_B}(a)\subseteq A\cap B$. Hence, $A\cap B$ is an $*$-open set. The rest of the proof is similar to the proof of Theorem \ref{topology}.
\end{proof}
Besides \ref{bl4} and \ref{b8}, the following theorem show that the introduced topologies $T_*$ and $T_\bigs$ on $L$ and $L^2$, makes $\mathcal{L}$ a topological BL-algebra.
\begin{theorem}
The mappings $*:(L^2,T_\bigs)\to(L,T_*)$ and $\too:(L^2,T_\bigs)\to(L,T_*)$ are continuous functions.
\end{theorem}
\begin{proof}
Consider an $*$-open set $A\in T_*$. We must show that the inverse images of $A$, $*^{-1}(A)$ and $\too^{-1}(A)$ are $\bigs$-open sets.

Firstly, consider a point $\mathbf{a}\in *^{-1}(A)$, that is $a_1 * a_2\in A$. Since $A$ is
an $*$-open set, there exists $r\ll 1$ such that
$B_r(a_1 * a_2)\subseteq A$. To finalize the first part of proof, we show that
$\mathbf{B}_r(\mathbf{a})\subseteq *^{-1}(A)$. To this end consider an element $\mathbf{b}\in\mathbf{B}_r(\mathbf{a})$. So, $\mathbf{a}\Leftrightarrow\mathbf{b}>r$, that is
\begin{equation}\label{a11}
(a_1\leftrightarrow b_1)*(a_2\leftrightarrow b_2)>r.
\end{equation}
On the other hand since by \ref{b15}
\begin{center}
$\medmuskip=1mu
\thinmuskip=1mu
\thickmuskip=1mu
(a_1 * a_2)\too (b_1 * b_2)\ge (a_1\too b_1)*(a_2\too b_2)$ and
$
\medmuskip=1mu
\thinmuskip=1mu
\thickmuskip=1mu
(b_1 * b_2)\too (a_1 * a_2)\ge (b_1\too a_1)*(b_2\too a_2)$,
\end{center}
so, applying \ref{b5} twice and then using \ref{b1} and \ref{a11} we get
\begin{eqnarray*}
(a_1 * a_2)\leftrightarrow(b_1 * b_2)&=&\big((a_1 * a_2)\too (b_1 * b_2)\big)
*\big((b_1 * b_2)\too (a_1 * a_2)\big)\\
&\ge&\big((a_1\too b_1)*(a_2\too b_2)\big)*\big((b_1 * b_2)\too (a_1 * a_2)\big)\\
&\ge&\big((a_1\too b_1)*(a_2\too b_2)\big)*\big((b_1\too a_1)*(b_2\too a_2)\big)\\
&=&\big((a_1\too b_1)*(b_1\too a_1)\big)*\big((a_2\too b_2)*(b_2\too a_2)\big)\\
&=&(a_1\leftrightarrow b_1)*(a_2\leftrightarrow b_2)\\
&>&r.
\end{eqnarray*}
Thus $b_1 * b_2\in B_r(a_1 * a_2)\subseteq A$. Hence $\mathbf{b}\in *^{-1}(A)$ which finalize the first part of the proof.

Secondly, to prove that $\too^{-1}(A)$ is an $\bigs$-open set, consider a point $\mathbf{a}\in \too^{-1}(A)$. So $a_1 \too a_2\in A$. Since $A$ is
an $*$-open set, there exists $r\ll 1$ such that
$B_r(a_1\too a_2)\subseteq A$. To prove that
$\too^{-1}(A)$ is $\bigs$-open, we show that
$\mathbf{B}_r(\mathbf{a})\subseteq \too^{-1}(A)$. To this end, if
$\mathbf{b}\in\mathbf{B}_r(\mathbf{a})$ ,then
\begin{equation}\label{a12}
\mathbf{a}\Leftrightarrow\mathbf{b}>r.
\end{equation}
By \ref{b9}
$a_1\too b_1\le (b_1\too b_2)\too(a_1\too b_2)$ and so by \ref{bl3}
\begin{equation}\label{a22}
(a_1\too b_1)*(b_1\too b_2)\le(a_1\too b_2).
\end{equation}
Again by \ref{b9}
$a_1\too b_2\le (b_2\too a_2)\too(a_1\too a_2)$ and so by \ref{a22}
\begin{center}
$(a_1\too b_1)*(b_1\too b_2)\le(b_2\too a_2)\too(a_1\too a_2)$.
\end{center}
Now applying \ref{bl3} we have
$\big((a_1\too b_1)*(b_1\too b_2)\big)*(b_2\too a_2)\le(a_1\too a_2)$
which besides \ref{b1} leads to
$\big((a_1\too b_1)*(b_2\too a_2)\big)*(b_1\too b_2)\le(a_1\too a_2)$.
Again \ref{bl3} implies that
\begin{equation}\label{a33}
(b_1\too b_2)\too(a_1\too a_2)\ge (a_1\too b_1)*(b_2\too a_2).
\end{equation}
A similar argument show that
\begin{equation}\label{a44}
(a_1\too a_2)\too(b_1\too b_2)\ge (b_1\too a_1)*(a_2\too b_2).
\end{equation}
Now by \ref{a33} and \ref{a44} and applying \ref{b5} we get
\begin{eqnarray*}
(a_1\too a_2)\leftrightarrow(b_1\too b_2)&=&
\big((a_1\too a_2)\too(b_1\too b_2)\big)*
\big((b_1\too b_2)\too(a_1\too a_2)\big)\\
&\ge&
\big((b_1\too a_1)*(a_2\too b_2)\big)*
\big((b_1\too b_2)\too(a_1\too a_2)\big)\\
&\ge&
\big((b_1\too a_1)*(a_2\too b_2)\big)*
\big((a_1\too b_1)*(b_2\too a_2)\big)\\
&=&
\big((b_1\too a_1)*(a_1\too b_1)\big)*
\big((a_2\too b_2)*(b_2\too a_2)\big)\\
&=&(a_1\leftrightarrow b_1)*(a_2\leftrightarrow b_2)\\
&=&\mathbf{a}\Leftrightarrow\mathbf{b}
\end{eqnarray*}
Thus by \ref{a12},
$(a_1\too a_2)\leftrightarrow(b_1\too b_2)>r$ that is $(b_1\too b_2)\in B_r(a_1\too a_2)\subseteq A$.
Hence $\mathbf{b}\in \too^{-1}(A)$ which completes the proof.
\end{proof}

\bibliographystyle{acm}

redpajama_set_name: RedPajamaArXiv


The Godalming High Street based social enterprise – part of The Meath Epilepsy Charity, which offers housing, care and support to people living with complex epilepsy and associated disabilities – has welcomed around 60 volunteers taking part in the build process, including people supported by the Meath, youngsters completing their Duke of Edinburgh Awards and people with mental health problems.
Once they are all completed, the boxes will be fixed to trees throughout the park in Godalming in the New Year.
The boxes are all crafted from soft wood, to RSPB standards and have been varnished with a natural look. They have different sized holes to attract a wide variety of birds.
Fiona Collier, Manager of Changing Perceptions, said: "Not only have the people we support, and the many volunteers involved, had some real fun making the boxes, but they are all gaining new skills and a great deal of self-confidence in knowing that their work will be much admired in the park and hopefully, will attract many more birds into the area.
Over the past five years The Phillips Memorial Park has been transformed by Waverley with the help of grants from the Heritage Lottery Fund and Big Lottery Fund and the bird boxes are the latest addition to the Green Flag Award-winning site. The project has led to the restoration of the Grade II listed Memorial Cloister, new signage, improved signposting, new planting, enhanced biodiversity and access improvements.
Councillor Kevin Deanus, Portfolio Holder for Operational Services, said: "One of our aims as part of The Phillips Memorial Park Improvement Project was to increase volunteering opportunities at the park for people of all ages, abilities and backgrounds.
"The bird box project has certainly done just that. I'm so impressed at how professional they look and by the dedication of the volunteers and everyone involved."


I'm trying out all my Designer Series polishes. Completely addicted to them. Gah, I loves me some sparkly. This is a brown polish with slight red undertones. There's also green shimmer in there, but I just can't capture it. Getting pictures of this polish was a massive pain. I just couldn't capture all the different colours. Definitely click the pictures to see the larger versions.
This was taken in artificial light.
This is three coats of colour. I totally screwed up on a finger and required the third coat. But it's good to go in two. When I look at this, I think green milk chocolate. Hahaha. Ya know... That makes me think of mint Aero. Mint+chocolate=greatest partnering ever. Hmmm, is there a black polish with green shimmer out there? I wonder what that'd look like.
Anyhow, yes, I really love the polish, but I'm not sure if it actually looks good on me. As with all the DS polishes, application was good, still love the pro wide brush, don't know how I ever lived without it.
I just got this colour a couple of weeks ago, and haven't tried it yet! BUUUUUUTTTT...seeing your photos of the polish, i will apply it in a couple of minutes. Haha. LOVE LOVE LOVE.
Lol, thanks! Hope you like it!


Our guest speaker is not yet selected but should be someone worth hearing.
SBR Annual Awards and Tire Rack Solo Trophies will conclude the evening.
Dinner will include a choice of Chicken Picatta or Beef Tips with all the delicious extras. We will enjoy a full selection of pies, of course. A gluten-free or veggie entree is also available on request using the coupon below.
Take a $10 banquet discount for every day you worked our Street Survival events, SBR Majors or Regional Road Races, or Lane events in 2017, up to $20 total.


text:
O.C. hockey mom charged with having sex with teen players
An Orange County hockey mother has surrendered herself in connection with charges of sexually assaulting two teenage boys.
Kathia Maria Davis, 44, a divorced mother of three, is accused of molesting a 13-year-old boy and having unlawful sex with a 16-year-old boy on her son's hockey team, prosecutors say. She is accused of plying them with alcohol and throwing parties, according to prosecutors. Both victims were associated with the Beach City Lightning, which plays in Anaheim and Laguna Niguel. Davis is due in court Friday.
Davis is accused of letting the 13-year-old boy in 2005 or 2006 drink alcohol in her home during a sleepover, prosecutors say. After her son and daughter went to sleep on the floor of the room they were all in, Davis is accused of kissing and fondling the 13-year-old boy, prosecutors say.
The 16-year-old was also given alcohol during visits to her Laguna Niguel home, prosecutors say. She is charged with having sex with the boy four times at her home in her bedroom between July 1, 2010, and July 31, 2011, prosecutors say. Prosecutors alleged that on some of those occasions, two of her children were present in the house.
The 16-year-old eventually told his mother about the alleged sexual contact and the family alerted Orange County Sheriff's detectives. Davis, 44, is charged with two felony counts of lewd acts on a child under 14 related to the younger boy and four felony counts of unlawful sexual intercourse with the 16-year-old and a sentencing enhancement allegation for substantial sexual conduct with a child, prosecutors say.
If convicted, she faces a maximum sentence of 12 years and eight months in state prison.
'Dear Santa' letters need volunteers, U.S. Postal Service says
Cellphones, not bad drivers, called top danger on California roads
High winds: Wires, poles, cables leave tangled mess in Temple City
--Richard Winton
Photo: Kathia Maria Davis Credit: Orange County Sheriff's Department