\documentclass{amsart}
\usepackage{amsmath,amsthm,amssymb,url}
\def\Dj{\mbox{\raise0.3ex\hbox{-}\kern-0.4em D}}

\title{Mal'cev characterizations of congruence meet-semidistributivity in locally finite varieties}

\author{Ralph McKenzie}
\author{Petar Markovi\'c}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{claim}[theorem]{Claim}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{example}[theorem]{Example}

\newcommand{\lb}{\langle}
\newcommand{\rb}{\rangle}
\newcommand{\m}[1]{{\mbox{\uppercase {\bf {#1}}}}}
\newcommand{\sm}[1]{{\mbox{\scriptsize {\uppercase {\bf {#1}}}}}}
\newcommand{\cn}[1]{{\rm Con}\: {\m{#1}}}
\newcommand{\vr}[1]{{\uppercase {\mathcal{#1}}}}
\newcommand{\Sg}[1]{{\rm Sg}^{\sm #1}}
\newcommand{\Cg}[1]{{\rm Cg}^{\sm #1}}

\begin{document}
\setcounter{section}{1}
\maketitle

These problems were posed on Tuesday, September 20th by Ralph McKenzie and Petar Markovi\'c.

The following result was proved by Marcin Kozik in 2009, appeared in the paper \cite{KKVW}, using the strong version of the bounded width theorem by Libor Barto from \cite{barto-collapsewidth}:

\begin{theorem}\label{thm1}
Let $\vr v$ be a locally finite variety. $\vr v$ is congruence meet-semidistributive iff there exist $\vr v$-terms $p(x,y,z)$ and $q(x,y,z,u)$ which are both weak near-unanimity terms in $\vr v$ and such that $\vr v\models p(x,x,y)\approx q(x,x,x,y)$.
\end{theorem}

A proof which first copies Kozik's almost verbatim and then uses a compactness argument can be devised to prove (see \cite{JMMM}):

\begin{theorem}\label{thm2}
Let $\vr v$ be a locally finite variety. $\vr v$ is congruence meet-semidistributive iff for all $k\geq 3$ there exist $\vr v$-terms $p_k(x_1,\ldots,x_k)$ which are weak near-unanimity terms in $\vr v$ and a $\vr v$-term $b(x,y)$ such that for all $k\geq 3$, $\vr v\models p_k(x,\ldots,x,y)\approx b(x,y)$.
\end{theorem}

\begin{problem}\label{prb1}
Is Theorem~\ref{thm2} still true if we add "and $\vr v\models b(x,b(x,y))\approx b(x,y)$" at the end of its statement (i.e. the requirement that all $p_k$ are special in the terminology of \cite{maroti-mckenzie})?
\end{problem}

One partial result is known from \cite{maroti-mckenzie}:

\begin{theorem}\label{thm3}
Let $\vr v$ be a locally finite variety. $\vr v$ is congruence meet-semidistributive iff there exists $n\geq 3$ such that for all $k\geq n$ there exist $\vr v$-terms $p_k(x_1,\ldots,x_k)$ which are weak near-unanimity terms in $\vr v$ and a $\vr v$-term $b(x,y)$ such that for all $k\geq n$, $\vr v\models p_k(x,\ldots,x,y)\approx b(x,y)\approx b(x,b(x,y))$.
\end{theorem}

Another strong Mal'cev characterization of the congruence meet-semi\-dis\-tri\-bu\-ti\-vi\-ty in locally finite varieties proved in \cite{JMMM} is

\begin{theorem}\label{thm4}
Let $\vr v$ be a locally finite variety. $\vr v$ is congruence meet-semidistributive iff there exists a $\vr v$-term $p(x,y,z,u)$ such that $\vr v \models p(x,x,x,x)\approx x$ and $\vr v\models p(x,x,x,y)\approx p(x,x,y,x)\approx p(x,y,x,x)\approx p(y,x,x,x)\approx p(x,x,y,y)\approx p(x,y,x,y)$ $\approx p(x,y,y,x)$.
\end{theorem}

This strong Mal'cev condition is pretty strong syntactically and it implies most other known Mal'cev characterizations of congruence meet-semi\-dis\-tri\-bu\-ti\-vi\-ty in locally finite varieties. However, a computer search identified two other strong Mal'cev conditions which are even stronger than the one mentioned in Theorem~\ref{thm4}, so they imply congruence meet-semidistributivity, but may be actually equivalent to it in all locally finite varieties. They are:

\begin{equation}\label{SM5}
\begin{array}{ccc}
t(x,x,x,x)\approx & x &\\
t(x,x,y,z)\approx & t(y,z,y,x)\approx & t(x,z,z,y)
\end{array}\tag{BD}
\end{equation}

\begin{equation}\label{SM6}
\begin{array}{ccc}
t(x,x,x,x)\approx & x &\\
t(x,x,y,z)\approx & t(y,x,z,x)\approx & t(y,z,x,y)
\end{array}\tag{LS}
\end{equation}

\begin{problem}\label{prb2}
Does every locally finite congruence meet-semidistributive variety satisfy the strong Mal'cev condition \eqref{SM5}? And how about \eqref{SM6}?
\end{problem}

Mikl\'os Mar\'oti solved Problem~\ref{prb2} on Friday, September 23rd by providing two counterexamples which are algebras with six elements. Both generate congruence meet-semidistributive varieties, but the first one fails the condition \eqref{SM5} while the other one fails the condition~\eqref{SM6}.

\begin{thebibliography}{9}
\bibitem{barto-collapsewidth}
	Barto, L.:
	The collapse of the bounded width hierarchy.
	J. Logic Comput. 
	{\bf 26} no. 3 (2016), 923--943.

\bibitem{JMMM} 
Jovanovi\'c, J., Markovi\'c, P., McKenzie, R., Moore, M.: Optimal strong Mal’cev conditions for congruence meet-semidistributivity in locally finite varieties, Algebra Universalis (in press), 21pp.

\bibitem{KKVW}
	Kozik, M., Krokhin, A., Valeriote, M. Willard, R.:
	Characterizations of several Maltsev conditions.
	Algebra Universalis
	{\bf 73} no. 3 (2015), 205–-224,

\bibitem{maroti-mckenzie} 
	Mar\'oti, M., McKenzie, R.:
	Existence theorems for weakly symmetric operations.
	Algebra Universalis
	{\bf 59} no. 3-4 (2008),
	463--489
\end{thebibliography}
\end{document}

