text:
\section{Introduction}
The hadronic vacuum polarization $\Pi(Q^2)$ is of great importance in precision tests of the Standard Model of particle physics. It enters, for instance, the running of the QED coupling constant. Additionally, it currently represents one of the dominant uncertainties in the Standard Model prediction of the anomalous magnetic moment of the muon $(g-2)_\mu$.\\%$a_\mu$.\\
The leading order hadronic contribution $a_\mu^{HLO}$ is accessible by computing the hadronic vacuum polarization function $\Pi(Q^2)$ and convoluting it with the an electro-weak kernel $K_E(Q^2,m_\mu)$ \cite{Jegerlehner:2009ry,gen0,gen1},
\begin{equation}\label{eq:gminus}
a_\mu^{HLO} = 4\alpha^2\int dQ^2 K_E(Q^2,m_\mu) \Big( \Pi(Q^2) - \Pi(0) \Big)~~.
\end{equation}
The leading hadronic contribution to the vacuum polarization $e^2\Pi(Q^2)$ in the spacelike domain can be expressed through the vector meson spectral function via a once-subtracted dispersion relation,
\begin{equation}\label{eq:disp}
\Big( \Pi(Q^2) - \Pi(0) \Big) = Q^2 ~\int_0^\infty~ds \frac{\rho(s)}{s(s+Q^2)} ~~.
\end{equation}
In the dispersive approach \cite{Jegerlehner:2009ry} one replaces the spectral function with the experimentally accessible $R(s)$-ratio by making use of the optical theorem.
On the lattice the problem of determining $\widehat{\Pi}(Q^2)=4\pi^2(\Pi(Q^2) - \Pi(0))$ can be approached both from the left and the right hand side of Eq.~\ref{eq:disp}.\\
In the following, we extend our analysis of the leading hadronic contribution to the anomalous magnetic moment of the muon using both the four-momentum method \cite{gen1, gen2, gen3, gen4, gen5, gen6, DellaMorte:2012cf} (i.e. by evaluating the lhs of Eq.~\ref{eq:disp}) and the recently introduced mixed representation method \cite{gen9,Francis:2013fzp,Bernecker:2011gh} (i.e. by evaluating the rhs of Eq.~\ref{eq:disp}).
We systematically compare both methods and monitor their approach towards the physical point. The preliminary results presented here are derived from local-conserved two-point lattice vector correlation functions, computed on a subset of light two-flavor ensembles made available to us through the CLS effort with pion masses as low as 190\,MeV.

\section{The four-momentum method to compute $\widehat{\Pi}(Q^2)$}
On a Euclidean lattice the vacuum polarization tensor can be defined as the four dimensional Fourier transform of the vector current-current correlation function:
\begin{equation}\label{eqn:fourcorr}
\Pi_{\mu\nu}(Q) \equiv \int d^4x \, e^{iQ\cdot x} \langle j_\mu(x) j_\nu(0)\rangle.
\end{equation}
Here $O(4)$ invariance and current conservation imply the tensor structure
\begin{equation}
\label{eqn:tensorstruct}
\Pi_{\mu\nu}(Q) =\big(Q_\mu Q_\nu -\delta_{\mu\nu}Q^2\big) \Pi(Q^2).
\end{equation}
We can therefore extract $\Pi(Q^2 \geq Q^2_{latt,min}(L))$ from lattice calculations.
However, computing $\widehat{\Pi}(Q^2)$ following this recipe, one is faced with the problem that the additive renormalization $\Pi(Q^2=0)$ is not directly available since the lowest available $Q^2_{min}=Q^2_{latt,min}(L)$ is dictated by the lattice discretization.
Consequently it has to be estimated using an extrapolation procedure $Q^2_{min} \rightarrow Q^2=0$, using e.g. a Pad\'e Ansatz \cite{gen6,DellaMorte:2012cf,gen7,gen8}. In addition the integrand of Eq.~\ref{eq:gminus} is strongly peaked around the mass of the lepton, and with the muon mass at $m_\mu\simeq105.65$\,MeV \cite{PDG}, this is generally below the lattice momentum resolution. As a consequence the resulting value of $a_\mu^{HLO}$ depends crucially on the correct extrapolation and therefore precision lattice data at low momentum $Q^2$ \cite{gen7, gen8}.

\section{The mixed representation method to compute $\widehat{\Pi}(Q^2)$}
In addition to the four-momentum method, in \cite{gen9,Francis:2013fzp,Bernecker:2011gh} a new method to compute $\widehat\Pi(Q^2)$ without the problem of having to estimate $\Pi(Q^2=0)$ was proposed. It has the advantage that it can be used to calculate any value of the virtuality $Q^2$. To this extent one approaches $\widehat{\Pi}(Q^2)$ from the right hand side of Eq.~\ref{eq:disp} by noting the electromagnetic spectral function $\rho(s)$ is directly linked to the lattice vector meson current-current correlator $\langle j_\mu(x) j_\nu(0)\rangle$ in the mixed time-momentum representation
\begin{equation}
G(x_0, \vec k) \stackrel{\mu=\nu}{=} \int d^3x\,e^{i\vec k \vec x} \langle J_\mu(x_0,\vec x) J_\nu(0) \rangle = \frac{1}{2} \int_0^\infty ds \sqrt{s}\rho(s)  e^{-\sqrt{s}|x_0|}~~.
\end{equation}
Exploiting this observation one arrives at an expression for $\widehat{\Pi}(Q^2)$ in terms of an integral over Euclidean time of the mixed representation correlator \cite{Francis:2013fzp}:
\begin{equation}\label{eq:mixrep}
4\pi^2\Big( \Pi(Q^2) - \Pi(0) \Big) = 4\pi^2\,\int_0^{\infty} dx_0\, G(x_0,\vec k=0)\, \Big[ x_0^2 - \frac{4}{Q^2}\sin^2(\frac{1}{2}Qx_0)\Big]~~.
\end{equation}
Additionally $a_\mu^{HLO}$ can be evaluated directly without having to take the intermediate step of calculating $\widehat{\Pi}(Q^2)$ by plugging Eq.~\ref{eq:mixrep} into Eq.~\ref{eq:gminus}, see \cite{Francis:2013fzp,Bernecker:2011gh}. However, for comparing both methods in the following, we find the extra step of computing $\widehat{\Pi}(Q^2)$ useful.\\
For a rigorous result, Eq.~\ref{eq:mixrep} must be integrated for all Euclidean times $t\rightarrow \infty$, a requirement that cannot be fulfilled on a finite lattice.
However, since the correlator drops exponentially with time, the integral can be truncated with only a small cost in accuracy, provided that the lattice data is precise enough for a large enough time separation. Here, we aim to estimate the large distance part of the integral in Eq.~\ref{eq:mixrep} by extrapolating the vector correlator to its asymptotic behavior.
Consequently the result for $a_\mu^{HLO}$ depends crucially on the knowledge of the long distance correlator or equivalently the low lying spectrum. In principle, the difficulty of extrapolating $Q^2_{min} \rightarrow Q^2=0$ in the four-momentum method has been replaced by precisely determining the large distance behavior of the mixed representation correlator.

\section{Numerical Setup}

In the following, we study the chiral behavior of  $a_\mu^{HLO}$ and $\widehat{\Pi}(Q^2)$ using both methods on dynamical gauge
configurations with two mass-degenerate quark flavors.  The gauge
action is the standard Wilson plaquette action \cite{Wilson:1974sk},
while the fermions were implemented via the O($a$) improved Wilson
discretization with non-perturbatively determined clover coefficient
$c_{\rm sw}$ \cite{Jansen:1998mx}.  The configurations
were generated within the CLS effort using algorithms
based on the DD-HMC\cite{CLScode} and MP-HMC packages\cite{mphmc}.
We calculated local-conserved
correlation functions using the same discretization and masses as in
the sea sector on a set of lattice ensembles with $\beta=5.30$ entailing a lattice spacing of $a =
0.0631(21)$\,fm~\cite{Capitani:2011fg} and pion masses ranging between $m_\pi=451$\,MeV down to $m_\pi=190$\,MeV, see Tab.~\ref{tab:par} for a list of lattice parameters.
Note, in the following, all correlation functions for strange quark masses are available as partially quenched, valence observables.

\begin{table}[h!]
\begin{center}
\begin{tabular}{cccccc}
    	\hline
    	Lattice size & $L$ $[\mathrm{fm}]$ &
    	$m_\pi$ $[\mathrm{MeV}]$ & $m_\pi L$ & $N_{meas}(N_{conf})$  & Label\\
    	\hline
    	$64 \times 32^3$ & $2.0$ & $451$ & $4.7$ & $4000(1000)$ &  E5 \\
    	$96 \times 48^3$ & $3.0$ & $324$ &  $5.0$ & $1200(300)$ &  F6 \\
    	$96 \times 48^3$ & $3.0$ & $277$ &  $4.2$ & $1000(250)$ &  F7 \\
    	$128 \times 64^3$ & $4.0$ & $190$ & $4.0$ & $820(205)$ &  G8  \\
    	\hline
\end{tabular}
\caption{Table of lattice parameters. To study the chiral behavior the set of ensembles used is fixed at $\beta=5.30$ and a lattice spacing of $a =
0.0631(21)$\,fm~\cite{Capitani:2011fg}. All correlators were calculated with four sources per configuration.}
\label{tab:par}
\end{center}
\end{table}

\section{Numerical Results}

\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{std.png}
\includegraphics[width=0.45\textwidth]{corr.png}
\caption{ Left: Updated results for $\Pi(Q^2)$ using the four-momentum method with $a =
0.0631(21)$fm and pion masses ranging between $m_\pi=451$\,MeV and $m_\pi=190$\,MeV. The insert shows the especially interesting low $Q^2$ region. Right: The mixed representation vector meson correlator on the same lattice ensembles. At $x_0/a=19$ we extend the lattice data by a single exponential, since the signal is lost beyond this distance at our current level of statistics.}
\label{fig:update}
\end{figure}

Updating our ongoing programme to compute $a_\mu^{HLO}$, in Fig.~\ref{fig:update} (left) we show the results of $\Pi(Q^2)$ obtained using the four-momentum method. The calculation closely follows the procedure presented in \cite{gen6,DellaMorte:2012cf}, as such local-conserved currents were used to compute $\Pi_{\mu\nu}(Q)$. Following \cite{DellaMorte:2012cf,Bedaque:2004bx}, twisted boundary conditions were used to increase the set of available lattice momenta by three twist angles. In addition propagators were computed on four maximally separated source positions on every configuration. The small errors of the results in Fig.~\ref{fig:update} (left) with pion masses from $m_\pi=451$\,MeV down to $m_\pi=190$\,MeV indicate that we are capable of reaching very high precision for all but the lowest $Q^2$ values in this way. Turning to Fig.~\ref{fig:update} (right), where we show the corresponding mixed representation correlation functions, this decrease in accuracy manifests itself by a rapidly deteriorating signal in $G(x_0)$ for $x_0/a\geq 19$ or $x_0\geq 1.1$\,fm. This is linked to a substantial contribution in the low $Q^2$ region from the low lying spectrum in the vector channel, including the two-pion state. Therefore, the large distance behavior of $G(x_0)$, the sharp increase of errors in the low $Q^2$ region, are a reflection of the lattice data not being accurate enough to capture exactly this part of the spectrum. As a consequence, we conclude, that model-independent, precision results for $\Pi(Q^2)$ require improvement in the lattice determination of the low lying vector spectrum. In the mixed representation method one possibility is to concentrate on the low lying masses of the correlation function by setting up a GEVP with additional interpolating operators.\\
Based on the phenomenological observation, that one expects large contributions to $a_\mu^{HLO}$ up to distances of $\sim 1.5$\,fm \cite{Bernecker:2011gh, Francis:2013fzp}, the results in Fig.~\ref{fig:update} (right) indicate truncating Eq.~\ref{eq:mixrep} at $x_0\simeq 1.1$\,fm, will not suffice to obtain a full result of $a_\mu^{HLO}$. For this reason we smoothly extend the correlation function at $x_0\simeq 1.1$fm by assuming the ground state is reached at this point and decays with a single exponential. Although more elaborate models can be used to extend the correlation function, our present data does not warrant the introduction of additional assumptions.
In the following we will use the extended correlator and updated $\Pi(Q^2)$ to compute $a_\mu^{HLO}$ and $\widehat{\Pi}(Q^2)$. Note that the determination of the long-distance behavior of the vector
correlator involves a fit before this step. The four-momentum method, on the
other hand, relies on fitting the $Q^2$ behavior close to $Q^2=0$.

\section{Comparing the four-momentum and mixed representation methods}

\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{combine_G8_tw.png}
\includegraphics[width=0.45\textwidth]{combine_F7_tw.png}
\caption{ Calculating the difference of the HVP obtained from the four-momentum method and $\widehat{\Pi}(Q^2)$ from the mixed representation method, we arrive at a measure to monitor the systematics in the two different analyses. The top panels show $\widehat{\Pi}(Q^2)$ obtained from the four-momentum (STD) and mixed representation (MRM) methods on the G8 (left) and F7 (right) lattice ensembles. The bottom shows the difference, i.e. $\Pi(0)$, for the $Q^2$ available in the four-momentum method.}
\label{fig:combine}
\end{figure}

\begin{figure}[t!]
\centering
\includegraphics[width=0.6\textwidth]{amu_nf2+1.png}
\caption{The chiral behavior of $a_\mu^{HLO}$ in $N_f=2+1_q$ QCD at $a=0.0631(21)$\,fm. The shown results are obtained using the four-momentum (STD) and mixed representation methods (MRM). Together they give an estimate of $a_\mu^{HLO}$ taking into account the different analysis systematics. The physical point and PDG \cite{PDG} values are given in black for reference.}
\label{fig:amu}
\end{figure}

The four-momentum and the mixed representation methods are based on different treatments of lattice data for the vector
correlator. If both methods are controlled, they should yield consistent
results for $a_\mu^{HLO}$ and $\widehat{\Pi}(Q^2)$. Any deviation should
arise from the different systematics of the analysis machinery only. Hence, by imposing
$\widehat{\Pi}_{STD}(Q^2)=\widehat{\Pi}_{MRM}(Q^2)$ we can compute the
additive renormalization $\Pi(0)$, which can then be compared to the result
from extrapolating $\widehat{\Pi}_{STD}$ to $Q^2=0$,
\begin{equation}\label{eq:pidiff}
\widehat{\Pi}_{STD}(Q^2) = \widehat{\Pi}_{MRM}(Q^2)
~~~\Rightarrow~~~ \Pi(0) = \Pi_{STD}(Q^2) -  \Big( \Pi(Q^2) - \Pi(0) \Big)_{MRM} ~~.
\end{equation}
As a result are able to monitor the systematics due to the different analyses, while at the same time determining $\Pi(0)$.
In Fig.~\ref{fig:combine} we show the results for $\widehat{\Pi}(Q^2)$ and $\Pi(0)$ obtained via Eq.~\ref{eq:pidiff} on the G8 (left) and F7 (right), i.e. $m_\pi=190$\,MeV and $m_\pi=277$\,MeV, lattice ensembles.
To determine $\widehat{\Pi}(Q^2)$ in the four-momentum method we use a Pad\'e fit and follow the procedure outlined in \cite{gen6,DellaMorte:2012cf}.\\
The top panels of Fig.~\ref{fig:combine} show $\widehat{\Pi}(Q^2)$ obtained from the four-momentum (STD) and mixed representation (MRM) methods. Since $Q^2$ is discrete in the four-momentum method, these results are shown as points, while those obtained using the mixed representation method are given as bands. The bottom panels show the result of Eq.~\ref{eq:pidiff} over $Q^2$ for those $Q^2=Q^2_{STD}$ dictated by the four-momentum results.
In the direct comparison of $\widehat{\Pi}(Q^2)$ the two methods show very good agreement. However, we find the results of Eq.~\ref{eq:pidiff} highlight the differences especially in the low $Q^2$ region, as the estimate of $\Pi(0)$ shows a decreasing trend around $Q^2\simeq 1\mathrm{GeV}^2$ with a sharp increase around the lowest available $Q^2$. Above $Q^2\simeq 2\mathrm{GeV}^2$ for G8 and $Q^2\simeq 4\mathrm{GeV}^2$ for F7 we furthermore observe a flat behavior of the results.
Since all deviations are within their respective errors, our results indicate a very good agreement of both methods, with systematics within the quoted errors.\\
Integrating our respective results for $\widehat{\Pi}(Q^2)$ via Eq.~\ref{eq:gminus} we are able to map out $a_\mu^{HLO}$ as it approaches the chiral limit using both methods.
This is shown in Fig.~\ref{fig:amu}, whereby we added the quenched strange quark contribution to the quoted numbers. Taking the results of both methods together we can estimate $a_\mu^{HLO}$ taking into account the different analysis systematics.

\section{Conclusions}

We presented results on the leading order hadronic contribution to the anomalous magnetic moment of the muon as it approaches the chiral limit using two different analysis methods in lattice QCD. Comparing the four-momentum and mixed representation methods we find they serve as independent cross-checks of each other, since they process equivalent data. As a consequence they serve as a means to estimate the systematic uncertainties inherent in any calculation
performed in finite volume. We highlighted that a precision result of $a_\mu^{HLO}$ requires accurate knowledge of the asymptotic behavior of the vector meson current-current correlation function and discussed how this can be systematically achieved in the mixed representation method. In the future this will enable a precision determination of $a_\mu^{HLO}$ and will allow for a straightforward inclusion of the disconnected contributions.

\acknowledgments
We are grateful to our colleagues within CLS for sharing the lattice ensembles used. We thank Dalibor Djukanovic and Christian Seiwerth for their technical support. This work was granted access to the HPC resources of the Gauss Center for Supercomputing at
Forschungzentrum J\"ulich, Germany, made available within the Distributed European Computing
Initiative by the PRACE-2IP, receiving funding from the European Community's Seventh Framework
Programme (FP7/2007-2013) under grant agreement RI-283493 (project PRA039).
The correlation functions were computed
on the dedicated QCD platforms ``Wilson''  at the Institute for Nuclear Physics,
University of Mainz, and ``Clover''  at the Helmholtz-Institut Mainz.
This work was supported by the \emph{Center for Computational Sciences}
as part of the Rhineland-Palatinate Research Initiative.


text:
\section{Introduction}\label{sec:intro}

Median algebras have been investigated by several authors in the context of distributive algebras, semilattices, graphs and hypergraphs
and in the framework of convex and metric spaces (see, e.g.,  \cite{Avann1961,Bandelt1983,Barthelemy83,Birkhoff1947,Evans82,Monjardet76,Monjardet80,Mulder79,Sholander1954,Slater80,Verheul93}).
A \emph{median algebra} is a structure $\alg{A}=\struc{A, \me}$ for a set $A$ and a ternary symmetric operation $\me\colon A^3\to A$, called \emph{median}, such that
\begin{gather*}
\me(x,x,y)=x,\\
\me(\me(x,y,z), t, u)=\me(x,\me(y,t,u), \me(z,t,u)).
\end{gather*}
In fact, axiom systems using only 4 variables are also known \cite{Bandelt1983,Isbell1980,Kol-Mar74}. For instance, in \cite{Kol-Mar74} it was shown that
$$\me(x,x,y)=x\quad \text{and} \quad \me(\me(x,w,z),y,z)=\me(\me(y,z,w),x,z)$$
suffice to axiomatize median algebras. In particular, it immediately follows that
$$\me(\me(x,y,z), y, z)=\me(x,y,z).$$

It is well known \cite{Sholander1954} that each element $a$ of a median algebra $\alg{A}$ gives rise to a \emph{median semilattice}
$\struc{A, \leq_a}$ where $\leq_a$ is given  by
\[
x \leq_a y \quad \iff \quad \me(a,x,y)=x.
\]
In this semilattice $a$ is the bottom element and the associated binary operation, denoted $\wedge_a$, is  defined by $x\wedge_a y =\me(a,x,y)$.
As shown in \cite{Sholander1954}, median semilattices coincide exactly with those $\wedge$-semilattices whose principal ideals
$$\downarrow x:=\{y\in A\colon y\leq_a x\}$$
are distributive lattices, and such that for any $a,b,c\in A$, $a\wedge b, b\wedge c, c\wedge a$ have a supremum whenever each pair of these meets is bounded above.
In such a case, we can define a median operation by
\begin{equation}\label{eqn:intro}
\me_\leq(x,y,z)=(x\wedge y)\vee (x\wedge z)\vee (z\wedge y),
\end{equation}
for every $x,y,z \in A$. Moreover, $\me=\me_{\leq_a}$ for every median algebra $\alg{A}=\struc{A, \me}$, and every $a\in A$ \cite{Avann1961}.
Similarly, every distributive lattice gives rise to a median algebra using \eqref{eqn:intro} and the converse also holds whenever there are $a,b\in A$ such that
$\me(a,x,b)=x$ for every $x \in A$.

Another noteworthy connection is  to \emph{median graphs}, i.e., connected graphs having the property that for any three vertices $a,b,c$,
there is exactly one vertex $x$ in the intersection of the shortest paths between any two vertices in $\{a,b,c\}$. As shown in  \cite{Avann1961},
the \emph{covering graph} (i.e., undirected {Hasse} diagram) of
every median semilattice whose intervals are finite is a median graph. Conversely,
every median graph is the covering graph of a median semilattice. For further background see, e.g.,  \cite{Bandelt1983}.

In \cite{CMT} the authors studied ``conservative'' median algebras, i.e.,  median algebras that satisfy $\me(x,y,z)\in \{x,y,z\},$ for all $ x, y, z \in A.$
Apart from the 4-element Boolean algebra, it was shown that conservative median algebras
can be represented by two lower bounded chains whose least elements are identified (thus, they can always be regarded as chains).
It is noteworthy to observe that they are exactly those median algebras whose subsets are themselves median (sub)algebras (in the terminology of \cite{BandeltVel99},
those with median stabilization degree equal to 0). Equivalently, they were shown to have the 4-element star as a forbidden subgraph.
The authors in \cite{CMT} also provided complete descriptions of median-homomorphisms between conservative median algebras (with at least 5 elements).

\begin{thm}[{\cite{CMT}}]\label{thm:1}
Let $\alg{A}$ and $\alg{B}$ be two conservative median algebras (thought of as chains) with at least five elements.
A map $f:\alg{A}\to \alg{B}$  is a median homomorphism if and only if $f$  is  monotone
(i.e., order-preserving or order-reversing).
\end{thm}

This result was then lifted, by making use of dual topological spaces of median algebras,  to median-homomorphisms between products of conservative median algebras.
For each positive integer $n$, we set $[n]=\{1, \ldots, n\}$.

\begin{thm}[{\cite{CMT}}]\label{thm:2}
Let $\alg{A}=\alg{C}_1\times  \cdots \times \alg{C}_n$ and  $\alg{B}=\alg{D}_1\times \cdots\times \alg{D}_k$ be two finite products of conservative median algebras (thought of as chains).
Then  $f:\alg{A}\to \alg{B}$ is a median homomorphism if and only if there exist $\sigma:[k]\to[n]$ and monotone maps
$f_i:\alg{C}_{\sigma(i)}\to \alg{D}_{(i)}$ for $i\in [k]$,
such that $f=(f_{\sigma(1)}, \ldots, f_{\sigma(n)})$.

In the particular case when  $k=1$,
\begin{equation*
f:\alg{C}_1\times \cdots \times \alg{C}_n\to \alg{D}
\end{equation*}
 is a median homomorphism
if and only if there is $j\in [n]$ and a monotone map $g:\alg{C}_j \to \alg{D}$ such that $f=g\circ \pi_j$.
\end{thm}

In this paper we are interested in median algebras $\alg{A}$ that are trees (i.e., acyclic undirected graphs) when thought of as semilattices:
a $\wedge$-semilattice (resp. $\vee$-semilattice) is said to be a \emph{tree} if no pair of incomparable elements have an upper (resp. lower) bound.
As we will see, such median algebras are obtained by relaxing
the condition of being conservative: we say that a median algebra $\alg{A}$ is a $(2:3)$-\emph{median semilattice} if
$\me(x,y,z)\in \{x\wedge y,y\wedge z,z\wedge x\}$ for all $ x, y, z \in A$.
Following the same structure as in \cite{CMT}, we will then proceed to describing the median homomorphisms between (products of) such median algebras.
As conservative median algebras with more than 4 elements  are particular cases of $(2:3)$-median semilattices, these descriptions properly extend results in \cite{CMT}.
Unlike in the latter paper, where an extensive use of dual spaces of median algebras took place, in the current paper we take a purely algebraic approach.

The paper is organized as follows. After recalling some terminology and basic results on median algebras in Section~\ref{Preliminary}, we introduce the notion of
$(2:3)$-median semilattice and show, in Section~\ref{sec:TreeMed}, that it captures those median semilattices that are trees. The problem of
describing homomorphisms between products of median algebras is then tackled in Section~\ref{sec:HomTree}. We show that such median preserving mappings can be decomposed into
 homomorphisms of the form
 \begin{equation}\label{eq:manymedto1}
  f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B},
 \end{equation}
 and explicitly describe them in the case when $\alg{B}$ is a tree.
  Our results are illustrated by several examples and their limitations by counter-examples.
 In Section~\ref{final} we comment on natural interpretations of our results, namely, when looked at as impossibility theorems.
 In particular, we show that Arrow type impossibility results for median-preserving proceedures \eqref{eq:manymedto1}
 (that basically state that they depend on at most one argument) can hold if and only
 if $\alg{B}$ is a tree.

\section{Preliminary results}\label{Preliminary}

In this section we introduce basic notions, terminology and notation, as well as
 recall some basic facts and well-known  results
about median algebras as ordered sets, that will be used throughout the paper. To avoid a lengthy preliminary section, we will restrict ourselves
to strictly necessary background, and we refer the reader to \cite{Birkhoff1948,Davey2002,Gratzer2003} for further background.

As discussed in Section~\ref{sec:intro}, when there is no danger of ambiguity,
we will not distinguish between median algebras and median semilattices. They will be denoted
by bold roman capital letters $\alg{A}, \alg{B}, \ldots$, while their universes will be denoted by  italic roman capital letters $A, B, \ldots$.
We will assume that universes of structures are nonempty, possibly infinite, sets.

Let $\alg{A}=\struc{A, \me}$ be a median algebra and let $a,b\in A$. The \emph{convex hull} of $\{a,b\}$ or the \emph{interval} from $a$ to $b$,
denoted by $[a,b]$, is defined by
$$[a,b]:=\{t\in A\colon t=\me (a,t,b)\}.$$
Intuitively, it is the set of all elements of $A$ in the ``shortest'' paths from $a$ to $b$, thus explaining our choice of notation.

\begin{prop}
 Let $\alg{A}=\struc{A, \me}$ be a median algebra and let $a,b\in A$. Then
 $$[a,b]=\{\me(a,t,b)\colon t\in A\}.$$
\end{prop}

Furthermore, we also have the following useful result.

\begin{prop}[{\cite{Birkhoff1947}}]\label{prop:2.2}
 Let $\alg{A}=\struc{A, \me}$ be a median algebra, and consider $a,b,c\in A$.
 Then $$[a,b]\cap[b,c]\cap[a,c]=\{\me(a,b,c)\}.$$
\end{prop}

 Let $\alg{A}=\struc{A, \me}$ be a median algebra and let $a,b\in A$.
 Each convex hull of a 2-element set $\{a,b\}$ can be endowed with a distributive lattice structure.
 To this purpose, for $a\in A$ recall the binary operation $\wedge_{a}\colon A^2\to A$ defined by:
$$x\wedge_{a}y:=\me (a,x,y),\quad \text{for }\, (x,y)\in A^2.$$
Note that such an operation is idempotent, commutative and associative.
Furthermore, it is not difficult to see that on $[a,b]$,  $\wedge_{a}$ and $\wedge_{b}$ verify the absortion laws, and that
for every $s\in [a,b]$, we have
$$a\leq_a s\leq_a b\qquad \text{and}\qquad  b\leq_b s\leq_b a.$$
In fact, we have the following well-known result; see, e.g., \cite{Bandelt1983}.

\begin{prop}\label{distributive-Interv}
 Let $\alg{A}=\struc{A, \me}$ be a median algebra. For every $a,b\in A$,
 $\struc{[a,b],\wedge_{a},\wedge_{b}}$ is a distributive lattice with $a$ and $b$ as the least and greatest elements, respectively.
\end{prop}

Moreover, we also have a general description of median-preserving mappings between two median algebras.

\begin{thm}\label{Hom-Median}
Let $\alg{A}$  and  $\alg{B}$ be median algebras, and consider
a mapping $f \colon A\to B$.
 Then the following assertions are equivalent.
\begin{itemize}

 \item[$(i)$] $f$ is a median-homomorphism.

 \item[$(ii)$] For all $p\in A$, $f \colon \struc{A, \wedge_{p}}\to \struc{B, \wedge_{f(p)}}$ is a semilattice-homomorphism.

 \item[$(iii)$] For all $p\in A$, $f \colon \struc{A, \leq_{p}}\to \struc{B,  \leq_{f(p)}}$ is an order-homomorphism.

 \item[$(iv)$]  For all $a,b\in A$, $f([a,b])\subseteq [f(a),f(b)]$.

\end{itemize}
\end{thm}

\begin{proof} We prove the following sequence of implications
$$(i)\implies (ii)\implies (iii)\implies (iv)\implies  (i)$$
thus showing that they are all equivalent.

$(i)\implies (ii)$: Suppose that $f$ is a median-homomorphism and $p\in A$. Then
$$f(a\wedge_{p}b)=f(\me(p,a,b))=\me(f(p),f(a),f(b))=f(a)\wedge_{f(p)}f(b).$$

$(ii)\implies (iii)$: Straightforward.

$(iii)\implies (iv)$: Let $t\in [a,b]$. Then $a\leq_a t\leq_a b$, and thus $f(a)\leq_{f(a)} f(t)\leq_{f(a)} f(b)$ from which it follows that $f(t)\in [f(a),f(b)]$.
In other words, $f([a,b])\subseteq [f(a),f(b)]$.

$(iv)\implies (i)$: Let $a,b,c\in A$. By Proposition~\ref{prop:2.2}, we have
$$\{\me(a,b,c)\}=[a,b]\cap[b,c]\cap[c,a].$$
By $(iv)$, we then conclude that
$$f(\me(a,b,c))\in [f(a),f(b)]\cap[f(b),f(c)]\cap[f(c),f(a)]=\{\me(f(a),f(b),f(c))\}.$$
Therefore, $f(\me(a,b,c))=\me(f(a),f(b),f(c)).$
\end{proof}

\begin{cor}\label{cor:new}
 Let $f\colon A\to B$ be a median homomorphism between two median algebras $\alg{A}$  and  $\alg{B}$. Then, for every $a,b\in A$,
 $f$ is also a lattice homomorphism from $\struc{[a,b],\wedge_{a},\wedge_{b}}$ to $\struc{[f(a),f(b)],\wedge_{f(a)},\wedge_{f(b)}}$.
\end{cor}

\section{Trees as median algebras}
\label{sec:TreeMed}

In this section we focus on median algebras that are trees.
In the particular case of chains, it was shown that these median algebras are exactly those that are conservative, i.e.,
$$\me(x,y,z)\in \{x,y,z\}, \quad x, y, z \in A.$$

To identify median semilattices that are trees, we propose the following relaxation of conservativeness.
We say that a median algebra, thought of as a median semilattice $\alg{A}=\struc{A, \wedge}$, is a $(2:3)$-\emph{median semilattice} if
for every $ x, y, z \in A$, we have
$$\me(x,y,z):=(x\wedge y)\vee (x\wedge z)\vee (z\wedge y)\in \{x\wedge y,y\wedge z,z\wedge x\}.$$

We start with a simple yet useful observation.

\begin{rem}\label{convexTreeChain}
 Let $\alg{A}=\struc{A, \me}$ be a median algebra such that $\struc{A, \leq_c}$ is a tree for some (or, equivalently, all) $c\in A$.
 Then, for all $a,b\in A$, the convex hull $[a,b]$  is a chain from $a$ to $b$.
\end{rem}

\begin{thm}\label{Median-Tree}
Let $\alg{A}=\struc{A, \me}$ be a median algebra. Then the following assertions are equivalent.
\begin{itemize}

 \item[$(i)$] There is $p\in A$ such that $\alg{A}=\struc{A, \leq_p}$ is a $(2:3)$-median semilattice.

 \item[$(ii)$] For all $p\in A$, $\alg{A}=\struc{A, \leq_p}$ is a $(2:3)$-median semilattice.

 \item[$(iii)$] There is $p\in A$ such that  $\alg{A}=\struc{A, \leq_p}$ is a tree.

 \item[$(iv)$]  For all $p\in A$,   $\alg{A}=\struc{A, \leq_p}$ is a tree.

 \item[$(v)$]  For all $a,b\in A$,  the bounded distributive lattice $\struc{[a,b],\wedge_{a},\wedge_{b}}$ is a chain.
\end{itemize}
\end{thm}

\begin{proof} We prove the following sequence of implications
$$(ii)\implies (i)\implies (iii)\implies (iv)\implies (v)\implies (iv)\implies (ii),$$
thus showing that they are all equivalent.

 $(ii)\implies (i)$: Straightforward.

 $(i)\implies (iii)$:  Suppose that $\alg{A}=\struc{A, \leq_c}$ is a $(2:3)$-median semilattice.
 To show that it is a tree it suffices to show that for every pair $a,b\in A$ with a common upper bound,
 we have $a \leq_p b$ or $b \leq_p a$.

 So suppose that $c\in A$ is a common upper bound of $a$ and $b$, that is, $\me(p,a,c)=a$ and $\me(p,b,c)=b$.
 Since $\alg{A}=\struc{A, \leq_c}$ is a $(2:3)$-median semilattice and $c$ is an upper bound of $a$ and $b$,
 \begin{equation}\label{eq:1}
  \me(c,a,b)\in \{\me(p,a,b), a, b\}.
 \end{equation}
Note that since $\me$ is a median,
$$\me(p,\me(a,c,b),a)=\me(b,\me(a,a,p),\me(c,a,p))=\me(b,a,a)=a,$$
and thus $a\leq_p \me(c,a,b)$. Similarly, we have $b\leq_p \me(c,a,b)$.

Now, if $ \me(c,a,b)=\me(p,a,b)$ in \eqref{eq:1}, then $\me(c,a,b)\leq_pa$ since $\me(p,a,b)\leq_pa$. Hence, $a=\me(c,a,b)$.
Similarly, we also have $b=\me(c,a,b)$, and thus $a=b$.

If $\me(c,a,b)=a$ in \eqref{eq:1}, then
$$\me(p,a,b)=\me(\me(c,a,b),p,b)=\me(a,\me(c,p,b),\me(b,p,b))=\me(a,b,b)=b,$$
and thus  $b \leq_p a$.
Similarly, if $\me(c,a,b)=b$ in \eqref{eq:1}, we conclude that $a \leq_p b$.

Since in all possible cases we have that $a \leq_p b$ or $b \leq_p a$,  $\alg{A}=\struc{A, \leq_p}$ is a tree.

 $(iii)\implies (iv)$: Straightforward.

 $(iv)\implies (v)$: Let $s,t\in [a,b]$. Thus $a\leq_a s,t\leq_a b$, and $s,t$ have a common upper bound in $\struc{A, \leq_a}$.  As $\struc{A, \leq_a}$ is a tree, $s$ and $t$
 cannot be incomparable, and hence the interval $[a,b]$ is a chain.

 $(v)\implies (iv)$: Let $a,b,p\in A$ and suppose that $a$ and $b$ are incomparable  in $\alg{A}=\struc{A, \leq_p}$. Note that $p\leq_p a,b$.

 For the sake of a contradiction, suppose that  $a$ and $b$  have a common upper bound $d\in A$, that is,
 $p\leq_p a,b\leq_pd$. However, by $(v)$ the interval $[p,d]$ is a chain and thus we must have $a \leq_p b$ or $b\leq_p a$, which constitutes the desired contradiction.

 $(iv)\implies (ii)$: Let $p\in A$. As $\alg{A}=\struc{A, \leq_p}$ is a tree, for every $a,b,c\in A$, we have that $a\wedge b, b\wedge c$ and $ c\wedge a$
 are pairwise comparable, and thus
 $$\me(a,b,c)\in \{a\wedge b,b\wedge c,c\wedge a\}.$$
 Hence, $\alg{A}=\struc{A, \leq_p}$ is a $(2:3)$-median semilattice.
\end{proof}

\begin{rem}\label{TreeChain} Other equivalent descriptions of trees as median semilattices are given in \cite{Bandelt1983,Sholander1954}.
\end{rem}

\begin{rem} As mentioned, conservative median semilattices $\mathbf{A}$ are $(2:3)$-median semilattices, whenever $|A|\geq 5$.
Hence, Lemma 3 and  Theorem 3 in \cite{CMT} follow as corollaries of Theorem \ref{Median-Tree}.
In fact, the conservative median semilattice $\{0,1\}^2$ is the only ``pathological'' case.
\end{rem}

\section{Median-homomorphisms over trees}\label{sec:HomTree}

Let $\alg{A}_1,  \cdots , \alg{A}_n$  and $\alg{B}_1,  \cdots , \alg{B}_k$ be median algebras. We seek to decsribe
median-homomorphisms of the form
\begin{equation}\label{eq:med-hom}
f\colon \alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}_1\times   \cdots \times  \alg{B}_k.
\end{equation}

For each $j\in [k]$, let $\pi_j$ denote the $j$-th projection on $\alg{B}_1\times   \cdots \times  \alg{B}_k$, i.e., the mapping
$$\pi_j\colon \alg{B}_1\times   \cdots \times  \alg{B}_k\to \alg{B}_j.$$
It is easy to see that if $f$ is a median-homomorphism, then
the composition
$$g_j=\pi_j\circ f\colon \alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}_j$$ is a median-homomorphism.
As the converse holds trivially, possibly with repeated $g_j$'s (as they are not necessarily pairwise distinct), we get the following result.

\begin{lemma}\label{lem:dec} Let $\alg{A}_1,  \cdots , \alg{A}_n$  and $\alg{B}_1,  \cdots , \alg{B}_k$ be median algebras.
 A mapping $f\colon \alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}_1\times   \cdots \times  \alg{B}_k$
 is a median-homomorphism if and only if there are  median-homomorphisms
 $$g_j\colon \alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}_j, \quad j\in [k],$$ such that
$f=(g_{1},\ldots, g_{k})$. Moreover, we have $g_j=\pi_j\circ f$ for the projection
$$\pi_j\colon \alg{B}_1\times   \cdots \times  \alg{B}_k\to \alg{B}_j,\quad j\in [k].$$
\end{lemma}

Hence the description of median-homomorphisms \eqref{eq:med-hom} boils down to describing  median-homorphisms of the form
$$ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}$$
from a finite product of median algebras $\alg{A}_1,  \cdots , \alg{A}_n$ (thought of as median semilattices)
to a median algebra $\alg{B}$. The general answer to this question still eludes us, but
we can provide explicit descriptions of such mappings when  $\alg{B}$ is of a special type.
We start by making some useful observations.

A function $ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}$
is called an \emph{$n$-median-homomrphism} if its ``unary sections'' $f_i\colon \alg{A}_i\to \alg{B}$ (obtained from $f$ by fixing all but the $i$-th argument)  are median-homomorphisms.
It is easy to verify that every median-homomorphism is an $n$-median-homomorphism.
However, the converse is not true. To see this, let $\alg{A}_1= \alg{A}_2= \alg{B}$ be the two element median algebra, and consider
$f\colon \alg{A}_1\times \alg{A}_2\to \alg{B}$ given by $f(1,1)=f(0,1)=f(1,0)=1$ and $f(0,0)=0$.
Clearly, $f$ is a 2-median homomorphism but it is not a median-homomorphism: for $\mathbf{a}=(0,0), \mathbf{b}=(0,1)$ and $\mathbf{c}=(1,0)$ we have
$$
f(\me(\mathbf{a},\mathbf{b},\mathbf{c}))=f(0,0)\neq \me(1,1,0)=\me(f(0,1),f(1,0),f(0,0)).
$$

Now we consider some particular cases, namely, when $\alg{B}$  is a chain (or, equivalently, a conservative median algebra) and
when it is a $(2:3)$-semilattice (or, equivalently, when $\alg{B}$ is a tree).

We start with the case when $\alg{B}$ (thought of as median semilattice) is a chain.

\begin{prop}\label{med-chain-homo}
 Let $\alg{A}_1,  \cdots , \alg{A}_n$ be median algebras and let $\alg{B}$  be a chain with $|B|\geq 2$.
 A mapping $ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}$ is a median-homomorphism if and only if
 there is an $i\in [n]$ and a median-homomorphism $g\colon  \alg{A}_i\to \alg{B}$ such that
 $$
 f(x_1,\ldots,x_n)=g(x_i),\quad \text{for all}\quad (x_1,\ldots,x_n)\in {A}_1\times \cdots \times {A}_n.
 $$
 In other words, $f$ is an essentially unary median-homomorphism.
\end{prop}

\begin{proof}

 Clearly, sufficiency holds. To prove necessity we show that $f$ cannot depend on two different arguments.

 For the sake of contradiction, suppose that $f$ depends on at least 2 of its arguments. Without loss of generality, we may assume that they are the first two.
 Hence, there are $a,b\in \alg{A}_1$ and $\mathbf{c}\in \alg{A}_1\times  \cdots \times \alg{A}_n$ such that $f(\mathbf{c}^a_1)>f(\mathbf{c}^b_1)$, where
 $\mathbf{c}^x_i$ denotes the tuple obtained from $\mathbf{c}$ by setting its $i$-th component to $x$.

 \begin{claim}\label{claim1}
  For every $\mathbf{d}\in \alg{A}_1\times  \cdots \times \alg{A}_n$, $f(\mathbf{d}^a_1)>f(\mathbf{d}^b_1)$.
 \end{claim}
\begin{proof}
Just consider the tuples $\mathbf{c}^a_1, \mathbf{c}^b_1,\mathbf{d}^a_1$ and $\mathbf{c}^a_1, \mathbf{c}^b_1,\mathbf{d}^b_1$,
and use the fact that $f$ is a median homomorphism.
\end{proof}

Similarly, there are  $a',b'\in \alg{A}_2$ and $\mathbf{c}'\in \alg{A}_1\times  \cdots \times \alg{A}_n$ such that $f({\mathbf{c}'}^{a'}_2)>f({\mathbf{c}'}^{b'}_2)$.
Again, we conclude that for every $\mathbf{d}\in \alg{A}_1\times  \cdots \times \alg{A}_n$, we have
$$f(\mathbf{d}^{a'}_2)>f(\mathbf{d}^{b'}_2).$$
Observe that, in particular, $f(\mathbf{d}^{a,a'}_{1,2})\geq f(\mathbf{d}^{b,a'}_{1,2}), f(\mathbf{d}^{a,b'}_{1,2}) \geq f(\mathbf{d}^{b,b'}_{1,2}) $.
As $\alg{B}$  is a chain, we must have
$$
f(\mathbf{d}^{b,a'}_{1,2})\geq f(\mathbf{d}^{a,b'}_{1,2}) \quad \text{or}\quad f(\mathbf{d}^{b,a'}_{1,2})\leq f(\mathbf{d}^{a,b'}_{1,2}).
$$
Without loss of generality, suppose that $\geq $ holds.

 \begin{claim}\label{claim2}
 If $f $ is a median homomorphism, then $f(\mathbf{d}^{a,a'}_{1,2})= f(\mathbf{d}^{b,a'}_{1,2})$.
 Similarly, If $f $ is a median homomorphism, then $f(\mathbf{d}^{a,b'}_{1,2}) = f(\mathbf{d}^{b,b'}_{1,2}) $.
 \end{claim}
\begin{proof}
We prove the first claim; the second follows analogously.
To see that we cannot have $>$, consider the tuples
$$f(\mathbf{d}^{a,a'}_{1,2})> f(\mathbf{d}^{b,a'}_{1,2})\geq f(\mathbf{d}^{a,b'}_{1,2})$$
to conclude that $f$ cannot then be a median-homomorphism.
\end{proof}

As Claim \ref{claim1} and  Claim \ref{claim2} contradict one another, the proof of Proposition~\ref{med-chain-homo} is now complete.
\end{proof}

The case when $\alg{B}$  is a $(2:3)$-semilattice (or, equivalently, when $\alg{B}$  is a tree), follows
from Proposition~\ref{med-chain-homo} by observing that if $a$ and $b$  are distinct elements of $\alg{B}$, then $[a,b]$ can be thought of as a chain (see Remark~\ref{convexTreeChain}).
In this case, we can reason as in the proof of Proposition~\ref{med-chain-homo} to obtain a more general result, namely,
when $\alg{B}$ is not necessarily a chain but a median algebra whose median graph is a tree.

\begin{figure}\label{figure3}
\begin{tikzpicture
\draw (0,0) node{$\bullet$}node[below]{$p=f(\mathbf{c}^{b,b'}_{1,2})$}--
(-1,1)node{$\bullet$}node[left]{$f(\mathbf{c}^{a,b'}_{1,2})$}--
(0,2)node{$\bullet$}node[above]{$f(\mathbf{c}^{a,a'}_{1,2})$} ;
\draw (0,0)--(1,1)node{$\bullet$}node[right]{$f(\mathbf{c}^{b,a'}_{1,2})$}--(0,2) ;
\end{tikzpicture}
 \caption{A square as in proof of Proposition~\ref{med-tree-homo}.}\label{fig:proof}
\end{figure}

\begin{prop}\label{med-tree-homo}
 Let $\alg{A}_1,  \cdots , \alg{A}_n$ be median algebras and let $\alg{B}$  be a tree.
 A mapping $ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}$ is a median-homomorphism if and only if
 there is an $i\in [n]$ and a median-homomorphism $g\colon  \alg{A}_i\to \alg{B}$ such that
 $$
 f(x_1,\ldots,x_n)=g(x_i),\qquad \text{for all}\quad (x_1,\ldots,x_n)\in {A}_1\times \cdots \times {A}_n.
 $$
 In other words, $f$ is an essentially unary median-homomorphism.
\end{prop}

\begin{proof} We provide an additional proof, alternative to that indicated in the paragraph preceding Proposition~\ref{med-tree-homo}.

As in the proof of Proposition \ref{med-chain-homo}, suppose that $f$ depends on at least 2 of its arguments. Without loss of generality, we may assume that they are the first two.
 Hence, there are $a,b\in \alg{A}_1$ and $\mathbf{c}\in \alg{A}_1\times  \cdots \times \alg{A}_n$ such that $f(\mathbf{c}^a_1)\neq f(\mathbf{c}^b_1)$.
 As in Claim \ref{claim1}, it is easy to see that the same holds for all $\mathbf{d}\in \alg{A}_1\times  \cdots \times \alg{A}_n$.

 Similarly, if the second argument of $f$ is essential, then are
 $a',b'\in \alg{A}_2$ and $\mathbf{c'}\in \alg{A}_1\times  \cdots \times \alg{A}_n$ such that $f(\mathbf{c'}^{a'}_2)\neq f(\mathbf{c'}^{b'}_2)$.
 Once again, we have in fact that $f(\mathbf{d'}^{a'}_2)\neq f(\mathbf{d'}^{b'}_2)$, for all $\mathbf{d'}\in \alg{A}_1\times  \cdots \times \alg{A}_n$.

 Hence, we have that  $f(\mathbf{c}^{x,y}_{1,2})$ for $(x,y)\in \{a,b\}\times\{a',b'\}$, are pairwise distinct. By picking $p$ of the form $f(\mathbf{c}^{x,y}_{1,2})$,
 it then follows that the four points $f(\mathbf{c}^{x,y}_{1,2})\in \alg{B}$ form a square as in Figure~\ref{figure3}.
 This contradicts the fact that  $\alg{B}$ is a tree.
\end{proof}

By Proposition~\ref{med-tree-homo}  to describe median-homomorphisms of the form
$$ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}$$ in the case when
$\alg{A}_1,  \cdots , \alg{A}_n$ are median algebras and $\alg{B}$ is a tree, it suffices to describe median homomorphisms $f\colon\alg{A}\to \alg{B}$ for a median algebra $\alg{A}$ and
a tree $\alg{B}$. Such descriptions follow from Theorem~\ref{Hom-Median}.

In the case when both $\alg{A}$ and  $\alg{B}$ are trees, Theorem~\ref{Hom-Median} together with Corollary \ref{cor:new} and Theorem \ref{Median-Tree}, imply the following proposition.

\begin{prop}\label{Hom-Median-Tree}
Suppose that the median algebras $\alg{A}$ and  $\alg{B}$ are trees, and consider
a mapping $f \colon A\to B$.
 Then the following assertions are equivalent.
\begin{itemize}

 \item[$(i)$] $f$ is a median-homomorphism.

 \item[$(ii)$] For all $a,b\in A$, the induced mapping $f \colon \struc{[a,b], \leq_{a}}\to \struc{[f(a),f(b)], \leq_{f(a)}}$ is an isotone function between chains.

\end{itemize}
\end{prop}

\begin{figure}

\begin{center}
\subfigure[$\alg{A}_1$]{\label{fig:bcv}\begin{tikzpicture}[scale=\tkzscl]
\draw (0,0) node{$\bullet$}node[below]{$a$}--(0,1)node{$\bullet$}node[right]{$b$};
\draw (0,1)--(-1,2)node{$\bullet$}node[left]{$c$};
\draw (0,1)--(1,2)node{$\bullet$}node[right]{$d$};
\end{tikzpicture}}
\hspace{2cm}
\subfigure[$\alg{A}_2$]{\label{fig:lat03_01} \begin{tikzpicture}[scale=\tkzscl]
\draw (0,2)--(0,2.75) node{$\bullet$}node[right]{$d'$};
\draw (0,0) node{$\bullet$} node[below]{$a$}--(-1,1)node{$\bullet$} node[left]{$b$};
\draw (0,0)--(1,1) node{$\bullet$} node[right]{$c$};
\draw (1,1)--(0,2) node{$\bullet$} node[right]{$d$};
\draw (-1,1)--(0,2);
\end{tikzpicture}}
 \end{center}

 \caption{Examples of $\wedge$-semilattices.}\label{fig:lat}
\end{figure}

In the case when $\alg{A}$ is  a tree and $\alg{B}$ is an arbitrary median algebra, Theorem~\ref{Hom-Median} can be strengthened by an additional and equivalent assertion, namely,
the existential variant of its assertion $(ii)$.

\begin{thm}\label{prop:added}
Let $\alg{A}$ be a tree and $\alg{B}$ an arbitrary median algebra. Then we have that one (or, equivalently, all) assertions in  Theorem~\ref{Hom-Median} are equivalent to
\begin{itemize}
\item[$(ii')$] There is a $p\in A$ such that $f \colon \struc{A, \wedge_{p}}\to \struc{B, \wedge_{f(p)}}$ is a semilattice-homomorphism.
\end{itemize}
\end{thm}

\begin{proof}
 We show that $(ii')\implies (i)$. In this way we have
 $$(ii)\implies (ii')\implies (i)\implies (ii)$$ and thus they are all equivalent.
 So suppose that $\struc{A,  \wedge_{p}}$ is a tree and that $$f \colon \struc{A, \wedge_p}\to \struc{B, \wedge_{f(p)}}$$ is a semilattice-homomorphism.
 In particular, for every $a,b,c\in A$, we have that
$$\me(a,b,c)\in \{a\wedge_p b,b\wedge_p c,c\wedge_p a\},$$
and that $f \colon \struc{A, \leq_{p}}\to \struc{B,  \leq_{f(p)}}$ is an order-homomorphism.

Without loss of generality, assume that $\me(a,b,c)=a\wedge_p b$. Hence,
$$a\wedge_p b=\me(a,b,c)=(a\wedge_p b)\vee_p(b\wedge_p c)\vee_p(c\wedge_p a)\geq_p b\wedge_p c,c\wedge_p a,$$
and, from the assumption that $f$ is  a semilattice-homomorphism,
\begin{eqnarray*}
f(\me(a,b,c))=f(a\wedge_p b)=f(a)\wedge_{f(p)}f( b)\geq_{f(p)} f(b)\wedge_{f(p)}f( c),f(c)\wedge_{f(p)}f( a).
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
&\me&(f(a),f(b),f(c))\\
&=&(f(a)\wedge_{f(p)}f( b))\vee_{f(p)} (f(b)\wedge_{f(p)}f( c))\vee_{f(p)}(f(c)\wedge_{f(p)}f( a)) \\
&=& f(a)\wedge_{f(p)}f( b)=  f(\me(a,b,c))
\end{eqnarray*}
and the proof of $(ii')\implies (i)$, and thus of the theorem, is now complete.
\end{proof}

\begin{rem}
 Note that the existential variant of $(iii)$ in Theorem~\ref{Hom-Median} is not equivalent to $(i)$ even in the case when $\alg{A}$ is a tree.
 To see this, consider the function on $\alg{A}_1$ (see Figure~\ref{fig:bcv})
 that maps $a,b$ to $a$, and leaves $c$ and $d$ fixed. Then it is an order-homomorphism for $p=a$, but it is not a median-homorphism.
\end{rem}

\begin{rem}
 Note also that Theorem~\ref{prop:added} does not necessarily hold in the case when  $\alg{A}$ is not a tree, even if $\alg{B}$ is conservative.
For instance, let $\alg{A}$ be the median algebra given in Figure~\ref{fig:lat03_01} and
 $\alg{B}$ the 4-element chain $1,2,3,4$, and consider the mapping
 that sends $d,d'$ to $4$, $b,c$ to $2$, and $a$ to $1$. Then, for all $p\in A$,
 $f \colon \struc{A, \wedge_{p}}\to \struc{B, \wedge_{f(p)}}$ is an order-homomorphism, but it is not a semilattice-homorphism for $p=a$
 (although it is a semilattice-homorphism for $p=d'$).
\end{rem}

\begin{problem} Given an arbitrary median algebra $\alg{B}$, describe those median algebras  $\alg{A}$ for which  Theorem~\ref{prop:added} holds.
\end{problem}

\section{Concluding remarks}\label{final}

In this paper we proposed
a natural relaxation
of conservativeness as considered in \cite{CMT}, which is of quite different flavour than that proposed in \cite{Bandelt1983,Sholander1954},
and showed that it axiomatizes the class of median algebras that are trees.
Moreover, we considered mappings
$$ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}_1\times\cdots \times \alg{B}_k$$
from products of arbitrary median algebras to products of $(2:3)$-semilattices (or, equivalently, trees)
and showed that those that preserve the median structure of  $\alg{A}_1\times  \cdots \times \alg{A}_n$ can be decomposed into
median-homomorphisms $g\colon \alg{A}_i\to \alg{B}_j$. The latter are described in Theorem~\ref{Hom-Median} and,
in the particular case when $\alg{A}_i$ is a tree, also in Proposition~\ref{Hom-Median-Tree} and in Theorem~\ref{prop:added}.

In the way, we looked into mappings
$$ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B},$$ where $\alg{A}_1,  \cdots , \alg{A}_n$ are arbitrary but where $\alg{B}$ is a $(2:3)$-semilattice, that is,  $\alg{B}$ is a tree.
The description of such median-homomorphims was then given in Proposition~\ref{med-tree-homo},  from which it follows that they are essentially unary.

Looking at them as aggregation proceedures that are not dictatorial (i.e., that depend on at least two arguments),
this translates into an impossibility theorem variant to that of Arrow \cite{Arrow50}.

Now the natural question is  to determine whether Proposition~\ref{med-tree-homo} still holds for an arbitrary median algebra  $\alg{B}$.
More precisely:

\begin{problem}\label{prob:last}
Given arbitrary median algebras  $\alg{A}_1,  \cdots , \alg{A}_n$,
 describe those median algebras  $\alg{B}$ for which all median-homomorphisms $ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}$ are trivial,
 i.e., essentially unary.
\end{problem}

Towards a solution to this problem, consider the following example.
Suppose that  $\alg{B}$ is an arbitrary median algebra thought as a $\wedge$-semilattice that is not a tree.
Thus, there are $a,b\in B$ with a nontrivial upper bound $c$, i.e., $c>a,b>a\wedge b$. Set $p=a\vee b$, $q=a\wedge b$. Note that, together with $a$ and $ b$, they form a square.
Now,  consider the median algebras $\alg{A}_1= \alg{A}_2=\{0,1\}$. Then $ f:\alg{A}_1\times  \alg{A}_2\to \alg{B}$ given by
$$f(1,1)=p,\quad f(0,1)=a,\quad f(1,0)=b,\quad f(0,0)=q$$
is a median homomorphism that depends on both of its variables. In other words, this is a counter-example to Problem~\ref{prob:last}.

Now this example can be easily extended to mappings $ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}$ and, as we have seen,
the condition that $\alg{B}$ is a  $(2:3)$-semilattice (i.e., a tree or, equivalently, does not contain a square as an order substructure) forces such
median-preserving mappings to be essentially unary.
From these considerations, we can thus provide an answer to  Problem~\ref{prob:last}, namely:
\begin{cor}
All median-homomorphisms $ f:\alg{A}_1\times  \cdots \times \alg{A}_n\to \alg{B}$ are essentially unary  if and only if $\alg{B}$ is a tree.
\end{cor}

redpajama_set_name: RedPajamaArXiv