\chapter{Preliminary Notions} \label{elementary} \label{centrality}

Here we fix notation and introduce definitions and results
with the purpose of providing a bridge from standard
material to the material in this monograph.

In this chapter and the sequel we use the following
conventions concerning notation.  The set of natural numbers
is $\omega$.  The first natural number is $0$.  Sequences or
tuples of elements from a set $X$ are written in boldface,
as in $\wec{x} \in X$ (or $\wec{x}\in X^n$ if the length $n$
is to be specified). The $i$-th coordinate of the tuple
$\wec{x}$ is written in italic as $x_i$.  If two sequences
$\wec{x}, \wec{y}\in X$ have the same length and $R$ is a
binary relation on $X$, then we may write
$\wec{x}\equiv\wec{y}\pmod{R}$ or $\wec{x}\equiv_R\wec{y}$
or $\wec{x}\wrel{R}\wec{y}$ to mean that $(x_i,y_i)\in R$
for all $i$.  If $R$ is a binary relation on $X$, then the
set $X^n[R]$ consists of the tuples $\wec{x}\in X^n$ such
that $x_i\equiv_R x_j$ for all $1\leq i\leq j\leq n$.  The
projection of a subset $S\subseteq \prod_{i\in I} X_i$ onto
a subset of coordinates $J\subseteq I$ is denoted
$\pi_J\colon S\to \prod_{j\in J} X_j$, and its kernel is
denoted $\eta_J$.  If $\theta$ is an equivalence relation on
$\prod_{j\in J} X_j$, then $\theta_J$~denotes
$\pi_J^{-1}(\theta)$ (so $0_J = \eta_J$).  When $J = \{j\}$,
then we write $\pi_j$,~$\eta_j$ and~$\theta_j$ instead of
$\pi_{\{j\}}$, $\eta_{\{j\}}$ and~$\theta_{\{j\}}$.
Expressions $A:= B$ or $B =: A$ mean ``$A$~is defined
by~$B$''.


\section{Algebras, Varieties, and Clones} \label{avc} 

An algebra is a model of a 1-sorted first-order algebraic
language.  To fix conventions, an {\bf algebraic
  signature}\index{signature} is a pair $\sigma:=(F,\alpha)$
where $F$ is a set (of operation symbols), and $\alpha\colon
F\to \omega$ is a function (assigning arity).  An {\bf
  algebra}\index{algebra} of signature $\sigma$ is a pair
$\m a:=\lb A; F\rb$ where $A$ is a nonempty set, called the
{\bf universe}\index{universe} of $\m a$, and for each $f\in
F$ with $\alpha(f)=k$ there is an assigned $k$-ary operation
$f^{\m a}\colon A^k\to A$.  The operations $f^{\m a}$ are
called the {\bf basic operations}\index{basic operations} of
$\m a$.

If $Y$ is a set that is disjoint from $F$, then $T(Y)$ is
the smallest set of words in the alphabet $F\cup Y$
satisfying
\begin{enumerate}
\item[(i)] $Y\subseteq T(Y)$, and
\item[(ii)] if $f\in F$, $\alpha(f)=k$ and
  $g_1,\ldots,g_k\in T(Y)$, then the word \hbox{$fg_1\cdots
    g_k\in T(Y)$.}
\end{enumerate}
When $Y = X := \{x_i\mid 1\leq i<\omega\}$, then $T(X)$ is
the set of {\bf terms of~signature $\sigma$}, and its
members are called {\bf terms}.\index{terms} Similarly, if
\hbox{$Y = X_n := \{x_1,\ldots,x_n\}$,} the members of
$T(X_n)$ are {\bf $n$-ary terms}.

The set $T(Y)$ has a canonical structure of an algebra of
signature~$\sigma$.  Namely, if $f\in F$ has arity $k$, then
in the term algebra $\m t = \m t(Y)$ the operation $f^{\m
  t}$ is defined so that if $g_1,\ldots,g_k\in T$, then
\hbox{$f^{\m t}(g_1,\ldots,g_k) = fg_1\cdots g_k\in T$.}  If
$\m a$ is an algebra of signature~$\sigma$ and $Y=X$ or
$X_n$, then an {\bf assignment in 
$\m a$}\index{assignment of variables} 
of the variables $Y$ is a function $v\colon
Y\to A$. It follows from the unique readability of terms
that any assignment~$v$ extends uniquely to an algebra
homomorphism $\widehat{v}\colon \m t\to \m a$; that is, $\m
t$ is free over $Y$ in the class of all algebras of
signature~$\sigma$.  If $t\in T$, then both $\widehat{v}(t)$
and $t(v)$ denote the image of~$t$ under~$\widehat{v}$.

If $\m a$ is an algebra of signature $\sigma$ and $t\in
T(X_n)$, then $t$ determines an $n$-ary {\bf term
  operation}\index{term operation} on $A$, the function
$t^{\m a}\colon A^n\to A$ is defined by
$(a_1,\ldots,a_n)\mapsto t(v)$ where $v\colon X_n\to A$ is
the assignment~\hbox{$x_i\mapsto a_i$.}  These~remarks apply
in the case where $\m a = \m t = \m t(X_n)$, and show that,
if $t\in T(X_k)$ and $g_1,\ldots, g_k\in T(X_n)$, then there
is a term $t(g_1,\ldots,g_k):=t^{\m t}(g_1,\ldots,g_k)\in
T(X_n)$.  It can be proved by induction that, in this
notation, the term $t(x_1,\ldots,x_k)$ is simply $t$ itself.

The {\bf language}\index{language} $L$ associated with the
signature $\sigma$ is the set of first-order formulas in
this signature.  Parentheses may be added to formulas to
improve readability.  An {\bf identity}\index{identity} in
the language~$L$ is an atomic $L$-formula, which is a
formula of the form $p\approx q$ where $p, q\in T(X_n)$ for
some~$n$.  An algebra $\m a$ {\bf
  satisfies}\index{satisfies!an identity} $p\approx q$,
written $\m a\models p\approx q$, if $p(v)=q(v)$ for every
assignment in $\m a$ (equivalently if $p^{\m a}=q^{\m a}$).
A~class of algebras in the language $L$ is a {\bf
  variety}\index{variety} if it is definable as the class of
all $L$-algebras satisfying some set of identities. The
smallest variety containing an algebra $\m a$ is denoted
${\vr V}(\m a)$.

A {\bf quasi-identity}\index{quasi-identity} is an open
first-order formula of the form
\[
\bigwedge_{i=1}^n (p_i\approx q_i) \to (p\approx q)
\]
where $p\approx q$ and each $p_i\approx q_i$ is an identity.
A class of $L$-algebras is a {\bf
  quasi-variety}\index{quasi-variety} if it is definable by
some set of quasi-identities.

A {\bf clone}\index{clone} is a multisorted structure ${\vr
  C} = \lb C_0, C_1, \ldots; F\rb$, with sorts indexed by
$\omega$, where
\begin{enumerate}
\item[(i)] Each of the sorts $C_i$, $i\in {\omega}$, is a
  set.
\item[(ii)] $F = \{\pi_i^n\mid n\in {\omega}, 1\leq
  i\leq n\} \cup \{{\rm comp}_n^m\mid m, n\in
  {\omega}\}$ is a set of operations between the $C_i$.
\item[(iii)] Each $\pi_i^n$ is a $0$-ary operation (a
  constant) in $C_n$.
\item[(iv)] ${\rm comp}_n^m\colon C_m\times
  \left(C_n\right)^m\to C_n$ is an $(m+1)$-ary operation.
\item[(v)] ${\rm comp}_n^n(f,\pi_1^n, \ldots, \pi_n^n) = f$
  and ${\rm comp}_n^m(\pi_i^m,f_1,\ldots,f_m) = f_i$.
\item[(vi)] ${\rm comp}_{m}^p\big(f,{\rm comp}_{m}^n(g_1,
  h_1,\ldots, h_n), \ldots, {\rm comp}_{m}^n(g_p,
  h_1,\ldots, h_n)\big)\!= {\rm comp}_{m}^n\big({\rm
    comp}_n^p(f,g_1, \ldots, g_p), h_1,\ldots, h_n\big)$.
\end{enumerate}
For an example of a clone, take $C_n$ to be the set of
$n$-ary $L$-terms, take $\pi_i^n$ to be $x_i$, and take
${\rm comp}_n^m$ to be the composition of an $m$-ary term
with $m$ $n$-ary terms.  This is the {\bf clone of
  $L$-terms}.  The 
{\bf clone of~$\m a$}\index{clone!of an algebra} 
when $\m a$ is an algebra is defined similarly
using the term operations of~$\m a$ instead of the
$L$-terms.

It is evident that clones are multisorted algebras defined
by identities, hence the class of clones is a multisorted
variety.  This implies that the usual algebraic notions
apply to clones. In particular, a sequence
\[
 {\bf h} = (h_0, h_1,\ldots )\colon {\vr C}\to {\vr D}
\]
is a {\bf homomorphism}\index{clone!homomorphism} between
clones if each $h_i\colon C_i\to D_i$ is a function, and the
sequence ${\bf h}$ preserves the clone operations of
composition and projection.  The notions of kernel and
quotient are defined in the obvious way.

Let $\vr V$ be a variety of 1-sorted algebras in the
language $L$.  The 
{\bf clone of $\vr V$},\index{clone!of a variety} 
denoted $\Clo({\vr V})$,\index{$\Clo$} is that
quotient of the clone of $L$-terms that is obtained by
identifying terms $p, q\in C_n$ if $p\approx q$ is satisfied
by all algebras in $\vr V$.  The correspondence $\vr
V\mapsto \Clo(\vr V)$ is essentially bijective. Namely, each
variety $\vr V$ is assigned the clone $\Clo(\vr V)$, and
each clone $\vr C$ is assigned a variety ${\rm Var}({\vr
  C})$\index{${\rm Var}$} which we define now.  The
operation symbols of ${\rm Var}({\vr C})$ are $\bigcup_{i\in
  {\omega}} C_i$ where the arity of $f\in C_n$ is defined to
be~$n$.  The identities defining ${\rm Var}({\vr C})$ are of
two types:
\begin{enumerate}
\item[(i)] projection identities: $\pi_i^n(x_1,\ldots,x_n)
  \approx x_i$ for each $1\leq i\leq n$, and
\item[(ii)] composition identities:
  \[
  f(x_1,\ldots,x_n)\approx
  g\big(h_1(x_1,\ldots,x_n),\ldots,h_m(x_1,\ldots,x_n)\big)
  \]
  whenever the equality $f = {\rm comp}_n^m(g, h_1,\ldots,
  h_m)$ holds in $\vr C$.
\end{enumerate}
It can be shown that $\vr C = \Clo\big({\rm Var}({\vr
  C})\big)$ holds for any clone $\vr C$, while $\vr V$ is
definitionally equivalent to the variety ${\rm
  Var}\big({\Clo(\vr V)}\big)$ for any variety~$\vr V$.

We write $\vr U\leq \vr V$ if there is a homomorphism ${\bf
  h}\colon \Clo(\vr U)\to \Clo(\vr V)$.  This notation is
used to express the fact that the algebras in $\vr V$ have
an underlying $\vr U$-structure.

\section{Lattice Theory}
\label{lattice_theory}

A {\bf partial lattice}\index{partial lattice} is a
structure $\m p=\lb P; \vee, \wedge, \leq\rb$ where $\lb P;
\leq\rb$ is a partially ordered set and $\vee$ and $\wedge$
are partial binary operations on~$P$ (called {\bf join} and
{\bf meet} respectively)\index{join}\index{meet} such that,
if $a\vee b$ is defined, then it is the least upper bound
of $a$ and $b$ in $\lb P; \leq\rb$, and if $a\wedge b$ is
defined, then it is the greatest lower bound of $a$ and $b$
in $\lb P; \leq\rb$.  A~\textbf{lattice}\index{lattice} is a
total algebra $\m l=\lb L; \vee, \wedge\rb$ on which there
is a partial order~$\le$ making $\lb L; \vee, \wedge,
\leq\rb$ a partial lattice.

If $x \leq y$ in some lattice $\m l$, then the {\bf
  interval}\index{interval} between them is $I[x,y] := \{
z\mid x\leq z \leq y\}$.  A {\bf lattice
  ideal}\index{ideal!of a lattice} of $\m l$ is a subset
$I\subseteq L$ closed under join ($x, y\in I\mimplies x\vee
y\in I$) and closed downward ($x\in I\wrel{\&} z\leq
x\mimplies z\in I$).  The dual concept is a {\bf lattice
  filter}.\index{filter} The {\bf principal
  ideal}\index{ideal!principal ---} determined by $x\in L$
is $(x]:= \{z\mid z\leq x\}$.  The collection $\vr I(\m
l)$\index{$\vr I(\m l)$} of all ideals of $\m l$, ordered by
inclusion, is the 
{\bf ideal lattice}\index{ideals!lattice of} 
of $\m l$.  If $I, J\in \vr I(\m l)$, then $I\wedge J =
I\cap J$ and
\[
I\vee J = \big\{z\mid \exists x\in I, y\in J (z\leq
x\vee y)\big\}\,.
\]

Lattice terms may be called 
{\bf words}.\index{lattice!--- words} 
If $p$ and $q$ are $n$-ary lattice words, then the
inclusion $p\leq q$ is {\bf satisfied} by a lattice $\m l$
if and only if $p^{\m l}\leq q^{\m l}$ in the pointwise
order.  Therefore $\m l$ satisfies $p\approx q$ if and only
if it satisfies both $p\leq q$ and $q\leq p$.  Note also
that $p\leq q$ is satisfied by~$\m l$ if and only if the
identity $p\approx p\wedge q$ is satisfied by~$\m l$. It
follows that a class of lattices is definable by identities
if and only if it is definable by inclusions. The variety of
all lattices is denoted 
$\vr L$.\index{$\vr L$ (the variety of lattices)}\index{L@$\vr L$ (the variety of lattices)}

If $Q$ is the lattice quasi-identity $\bigwedge (p_i\approx
q_i)\to (p\approx q)$, then the {\bf
  $Q$-configuration}\index{$Q$-configuration}\index{Q-configuration@$Q$-configuration}\index{configuration!$Q$--}
is a pair $\big(\m p(Q),\, p\approx q\big)$ where $\m p(Q)$
is the natural partial lattice of subterms of terms
of~$Q$. That is, $\m p(Q)$ is the partial lattice presented
by $\lb G \mid R\rb$ where $G$ is the set of subterms of
terms appearing in $Q$ and $R$ consists of relations of the
following types: if $s$,~$t$ and $s\vee t$ are subterms,
then $R$ contains a relation expressing that $s\vee t$ is
equal to the join of $s$ and $t$, $R$ contains similar
relations for meet, and for each premise $p_i\approx q_i$ of
$Q$ the set $R$ contains the relation $p_i=q_i$.  It is not
assumed that $R$ contains $p=q$ where $p\approx q$ is the
conclusion of $Q$.  This definition exists to make the
following statement true: an assignment of the variables of
$Q$ in a lattice $\m l$ which satisfies the premises of $Q$
determines and is determined by a homomorphism of partial
lattices $\varphi\colon\m p(Q)\to \m l$, and the assignment
will satisfy $Q$ precisely when $\varphi(p)=\varphi(q)$.

\begin{df}\label{sdm_df}
  The {\bf meet semidistributive 
    law}\index{meet semidistributive law} 
  is the quasi-identity
  \begin{equation}\label{sdm_law}
    \big((p\wedge q) \approx s\big)\wrel{\&}
    \big((p\wedge r) \approx s\big)\to
    \big((p\wedge (q \vee r)) \approx s\big)\,,
  \end{equation}
  and the 
  {\bf join semidistributive 
    law}\index{join semidistributive law} 
  is the dual quasi-identity.
\end{df}

Quasi-identity (\ref{sdm_law}) is equivalent to
\begin{equation}\label{sdm_equiv_version}
  \big((p\wedge q)\approx s\big)\wrel{\&}
  \big((p\wedge r)\approx s\big)\wrel{\&}
  \big((q\vee r)\geq p\big)\to
  (p\approx s)\,.
\end{equation}
The reason this is so is that any assignment of variables
which fails to satisfy (\ref{sdm_equiv_version}) will also
fail to satisfy (\ref{sdm_law}), while conversely if
$p\mapsto a$, $q\mapsto b$, $r\mapsto c$, $s\mapsto d$ is an
assignment of variables in some lattice that fails to
satisfy (\ref{sdm_law}), then $p\mapsto a\wedge (b\vee c)$,
$q\mapsto b$, $r\mapsto c$, $s\mapsto d$ is an assignment
that fails to satisfy (\ref{sdm_equiv_version}).

{\bf The
  $\sdm$-configuration}\index{$\sdm$-configuration}\index{configuration!$\sdj$
  or $\sdm$} is the $Q$-configuration where $Q$ is
quasi-identity (\ref{sdm_equiv_version}).  More explicitly,
let $\m p(\sdm)$ be the partial lattice generated by $\{p,
q, r, m, j\}$ where $m = p\wedge q = p\wedge r$ and $j =
q\vee r\geq p$.  Then the $\sdm$-configuration is $\big(\m
p(\sdm),\, p\approx m\big)$.  An {\bf
  $\sdm$-failure}\index{SDM@$\sdm$-failure (or $\sdj$-failure)}\index{failure!meet semidistributive or join semidistributive} 
in a lattice~$\m l$ is an interval of the form
$I=I[\varphi(m),\varphi(p)]$ where \hbox{$\varphi\colon \m
  p(\sdm)\to \m l$} is a homomorphism of partial lattices.
This $\sdm$-failure is {\bf trivial} if $I$ has one element
and is {\bf nontrivial} otherwise.  {\bf The
  \hbox{$\sdj$-configuration}} and {\bf $\sdj$-failures} are
defined dually.  Thus a lattice is meet semidistributive if
and only if it has no nontrivial $\sdm$-failures.

For finite lattices, or for varieties of lattices, the meet
and join semidistributive laws are characterized in the
following theorem.

\begin{thm} \label{not_jonsson_rival} \mbox{}
  \begin{enumerate}
  \item[(1)] {\upshape(Cf.\ \cite{davey-p-r})} A finite
    lattice is meet semidistributive if and only if it has
    no sublattice isomorphic to $\m m_3$, $\m d_1$, $\m
    e_1$, $\m e_2$ and\/ $\m g$.  (These are five of the
    following six lattices.)
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{.9mm}
\begin{picture}(115,35)
\put(0,20){\circle*{1.2}}
\put(15,5){\circle*{1.2}}
\put(15,20){\circle*{1.2}}
\put(15,35){\circle*{1.2}}
\put(30,20){\circle*{1.2}}
%
\put(45,15){\circle*{1.2}}
\put(45,25){\circle*{1.2}}
\put(60,5){\circle*{1.2}}
\put(60,15){\circle*{1.2}}
\put(60,35){\circle*{1.2}}
\put(75,15){\circle*{1.2}}
\put(75,25){\circle*{1.2}}
%
\put(90,15){\circle*{1.2}}
\put(90,25){\circle*{1.2}}
\put(105,5){\circle*{1.2}}
\put(105,25){\circle*{1.2}}
\put(105,35){\circle*{1.2}}
\put(120,15){\circle*{1.2}}
\put(120,25){\circle*{1.2}}
%
\put(15,5){\line(-1,1){15}}
\put(15,5){\line(0,1){15}}
\put(15,5){\line(1,1){15}}
\put(15,35){\line(-1,-1){15}}
\put(15,35){\line(0,-1){15}}
\put(15,35){\line(1,-1){15}}
%
\put(45,15){\line(3,-2){15}}
\put(45,15){\line(0,1){10}}
\put(60,15){\line(-3,2){15}}
\put(60,15){\line(3,2){15}}
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\put(60,35){\line(3,-2){15}}
\put(75,15){\line(-3,-2){15}}
\put(75,15){\line(0,1){10}}
%
\put(90,15){\line(3,-2){15}}
\put(90,15){\line(0,1){10}}
\put(105,25){\line(-3,-2){15}}
\put(105,25){\line(3,-2){15}}
\put(105,25){\line(0,1){10}}
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\put(105,35){\line(3,-2){15}}
\put(120,15){\line(-3,-2){15}}
\put(120,15){\line(0,1){10}}
%
\put(13,-1){$\m m_3$}\index{$\m m_3$}
\put(58,-1){$\m d_1$}\index{$\m d_1$}
\put(103,-1){$\m d_2$}\index{$\m d_2$}
\end{picture}
\end{center}

\bigskip\bigskip

\begin{center}
\setlength{\unitlength}{.9mm}
\begin{picture}(115,33)
\put(5,20){\circle*{1.2}}
\put(15,5){\circle*{1.2}}
\put(15,15){\circle*{1.2}}
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\put(25,15){\circle*{1.2}}
\put(25,25){\circle*{1.2}}
%
\put(50,20){\circle*{1.2}}
\put(60,5){\circle*{1.2}}
\put(60,25){\circle*{1.2}}
\put(60,35){\circle*{1.2}}
\put(70,15){\circle*{1.2}}
\put(70,25){\circle*{1.2}}
%
\put(90,15){\circle*{1.2}}
\put(100,5){\circle*{1.2}}
\put(100,25){\circle*{1.2}}
\put(105,20){\circle*{1.2}}
\put(110,15){\circle*{1.2}}
\put(110,35){\circle*{1.2}}
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%
\put(15,5){\line(-2,3){10}}
\put(15,5){\line(0,1){10}}
\put(15,5){\line(1,1){10}}
\put(15,35){\line(-2,-3){10}}
\put(15,35){\line(1,-1){10}}
\put(25,25){\line(-1,-1){10}}
\put(25,25){\line(0,-1){10}}
%
\put(60,5){\line(-2,3){10}}
\put(60,35){\line(0,-1){10}}
\put(60,5){\line(1,1){10}}
\put(60,35){\line(-2,-3){10}}
\put(60,35){\line(1,-1){10}}
\put(70,15){\line(-1,1){10}}
\put(70,25){\line(0,-1){10}}
%
\put(90,15){\line(1,1){20}}
\put(90,15){\line(1,-1){10}}
\put(100,25){\line(1,-1){10}}
\put(120,25){\line(-1,1){10}}
\put(120,25){\line(-1,-1){20}}
%
\put(13,-1){$\m e_1$}\index{$\m e_1$}
\put(58,-1){$\m e_2$}\index{$\m e_2$}
\put(104,-1){$\m g$}\index{$\m g$}
\end{picture}
\end{center}
\caption{ \ }\label{fig1}
\end{figure}
\item[(2)] {\upshape(Cf.\ \cite{jonsson-rival})} A variety
  of lattices consists of meet semidistributive lattices if
  and only if it does not contain $\m m_3$, $\m d_1$, $\m
  e_1$, $\m e_2$ or\/~$\m g$.
\end{enumerate}
The dual results hold for join semidistributivity in place
of meet semidistributivity.
\end{thm}

\begin{df}\label{(W)_df}
  {\bf Whitman's condition},\index{Whitman's condition (W)}
  denoted by ${\bf (W)}$,\index{(W)} is the formula
  \begin{align*}
  (x\wedge y\leq u\vee v&)\to\\
  \big((x\leq u\vee v&) \;{\rm
    or}\; (y\leq u\vee v) \;{\rm or}\; (x\wedge y\leq u)
  \;{\rm or}\; (x\wedge y\leq v)\big).
  \end{align*}
\end{df}

This condition may be written a little more compactly as:
\[
x\wedge y\leq u\vee v\quad\mbox{\rm  implies}\quad
\{x,y,u,v\}\cap I[x\wedge y,u\vee z] \neq \emptyset.
\]
All lattices depicted in Theorem~\ref{not_jonsson_rival}
satisfy $\W$.

\begin{df}\label{quasi_sat_W}
  A quasi-identity $ \bigwedge_{i=1}^n (p_i\approx q_i) \to
  (p\approx q) $ in $k$~variables {\bf satisfies (W)} if the
  lattice with the presentation \hbox{$\lb G \mid R\rb$}
  satisfies $\W$, where $G = \{g_1,\ldots, g_k\}$, $\wec{g}
  = (g_1,\ldots,g_k)$ and $R = \{p_1(\wec{g}) =
  q_1(\wec{g}),\ldots, p_n(\wec{g}) = q_n(\wec{g})\}$.
\end{df}

Note that whether or not a quasi-identity satisfies $\W$
depends only on its premises.  It is proved in
\cite{czedli-day} that a quasi-identity $Q$ satisfies $\W$
if and only if the associated partial lattice $\m p(Q)$
satisfies $\W$.  Since $\m p(Q)$ is finite, this yields an
algorithm for testing a quasi-identity for~$\W$.  It is
noted in \cite{czedli-day} that the meet semidistributive
law satisfies~$\W$, and that more generally any
quasi-identity whose premises are free of joins (as in form
(\ref{sdm_law}) of the meet semidistributive law)
satisfies~$\W$.  These remarks also apply to the join
semidistributive law.

\begin{df} \label{bounded} A surjective lattice homomorphism
  $h\colon \m k\to \m l$ is {\bf upper
    bounded}\index{bounded!upper} if each kernel class
  $h^{-1}(a)$, $a\in L$, has a largest element, is {\bf
    lower bounded}\index{bounded!lower} if each kernel class
  has a least element, and is {\bf bounded}\index{bounded}
  if it is both lower and upper bounded.  A lattice $\m l$
  is {\bf (upper, lower) bounded} if there is a surjective
  (upper, lower) bounded homomorphism $h\colon \m f_{\vr
    L}(x_1,\ldots,x_n)\to \m l$ from a finitely generated
  free lattice onto $\m l$.
\end{df}

Any lower bounded lattice is join semidistributive
(Theorem~2.20 of~\cite{freese-jezek-nation}).  Chapter~2 of
\cite{freese-jezek-nation} describes an effective algorithm
for testing if a finite lattice is lower or upper
bounded. We shall not have cause to use this algorithm, but
for later reference we point out that $\m d_1$ is the only
lower bounded lattice depicted in
Theorem~\ref{not_jonsson_rival}.  The others are excluded
because they are not join semidistributive, while the lower
boundedness of $\m d_1$ is proved one way in Example~2.74 of
\cite{freese-jezek-nation} and differently in Table~3 of
\cite{mckenzie1}.

\begin{df}\label{projective&splitting}
  An algebra $\m p$ is {\bf projective}\index{projective}
  relative to a variety~$\vr V$ if whenever $\sigma\colon \m
  a\to \m b$ is a surjective homomorphism between algebras
  of\/~$\vr V$ and $\varphi\colon \m p\to \m b$ is a
  homomorphism, then there is a homomorphism
  $\overline{\varphi}\colon \m p\to \m a$ such that
  $\sigma\circ \overline{\varphi} = \varphi$.
  \[
  {\xymatrix{
      & {\m p}\ar@{-->}[ld]_*+{\overline{\varphi}}\ar[d]^*+{\varphi}\\
      {\m A} \ar@{->>}[r]_*+{\sigma} & {\m B} }}
  \]
\end{df}

When $\vr V$ is a variety of lattices, we may call a partial
lattice $\m p$ 
{\bf projective}\index{projective!(partial) lattice} 
relative to $\vr V$ if it satisfies the
homomorphism lifting property of this definition.

\begin{thm}\label{proj+si_implies_splitting}
  If\/ $\vr V$ is a variety of algebras and $\m p$ is
  subdirectly irreducible and projective relative to $\vr
  V$, then there is an identity $\varepsilon$ such that for
  all $\m a\in {\vr V}$ it is the case that $\m a\models
  \varepsilon$ if and only if $\m a$ has no subalgebra
  isomorphic to $\m p$.
\end{thm}

We call the identity $\varepsilon$ the {\bf conjugate
  identity}\index{conjugate identity} for~$\m p$.

\begin{proof}
  Let ${\vr U}$ consist of all members of $\vr V$ that have
  no subalgebra isomorphic to $\m p$.  Then $\vr U$ is
  closed under the formation of homomorphic images because
  $\m p$ is projective relative to $\vr V$, $\vr U$ is
  closed under subalgebras by definition, and $\vr U$ is
  closed under products because $\m p$ is subdirectly
  irreducible.  By Birkhoff's $\Ho\Su\Pd$
  Theorem,\index{HSP~Theorem} $\vr U$ is a subvariety
  of~$\vr V$. Since $\m p\notin {\vr U}$, there is an
  identity $\varepsilon$ that is satisfied in $\vr U$ but
  not by $\m p$. Necessarily $\varepsilon$ fails in every
  member of ${\vr V}-{\vr U}$, so this identity has the
  property stated in the theorem.
\end{proof}

We shall have occasion to refer to the following
closely-related result.

\begin{thm}\label{proj+si_implies_splitting2}
  If\/ $Q$ is a quasi-identity and\/ $\m p(Q)$ is projective
  relative to ${\vr V}$, then the class of lattices in ${\vr
    V}$ that satisfiy $Q$ is a subvariety of\/~$\vr V$.
\end{thm}

\begin{proof}
  As above, let ${\vr U}$ consist of all members of $\vr V$
  that satisfy~$Q$.  Then $\vr U$ is closed under the
  formation of homomorphic images because $\m p(Q)$ is
  projective relative to $\vr V$, $\vr U$ is closed under
  subalgebras and products because $Q$ is a quasi-identity.
\end{proof}

\section{Meet Continuous Lattice Theory}\label{mc_lattice_theory}

A lattice is {\bf meet continuous}\index{lattice!meet continuous ---}\index{meet continuous lattice} 
if it is
complete and the binary meet operation distributes over
arbitrary up-directed suprema. That is, for every
up-directed set $\vr D$,
\begin{equation}\label{meet_cont_dist}
x\wedge \bigvee_{y\in \vr D} y = \bigvee_{y\in \vr D} x\wedge y\,.
\end{equation}
Every algebraic lattice
is meet continuous (Lemma VIII.5.2 of \cite{birkhoff}).

The language of meet continuous lattices has operation
symbols $\{\bigvee, \wedge\}$ where $\bigvee$ is a class of
$\kappa$-ary join operations for all $\kappa\geq 0$ and
$\wedge$ is binary meet.  The $0$-ary join is a constant that
interprets as the least element and is denoted $0$.  The
fact that $\wedge$ is a meet operation is expressible by
identities in the same way that it is expressed in lattice
theory.  As with binary join, it is easy to express with
identities the fact that $\kappa$-ary $\bigvee$ is
$\kappa$-ary join with respect to the $\wedge$-order.  To
show that meet continuity is expressible by identities one
must show that the distributive law in
(\ref{meet_cont_dist}) can be rewritten with quantification
over arbitrary sets rather than up-directed sets.  This is
done by replacing an arbitrary join with the up-directed
supremum of its finite sub-joins:
\[
x\wedge \bigvee_{y\in {\vr F}} y 
\wrel{\approx}
\bigvee_{
\begin{array}{c}
\mbox{\scriptsize $\vr F_0\subseteq \vr F$}\\
\mbox{\scriptsize $\vr F_0 \;\mbox{\rm finite}$}
\end{array}} 
\bigg(x\wedge \bigvee_{y\in {\vr F}_0} y\bigg).
\]

The category 
$\vr L_{MC}$\index{$\vr L_{MC}$ (the variety of meet continuous lattices)}\index{LAAMC@$\vr L_{MC}$ (the variety of meet continuous lattices)} 
of meet continuous
lattices with join complete homomorphisms has a forgetful
functor $F\colon \vr L_{MC}\to \vr L$ to lattices that
forgets all join operations except the binary
join. Theorem~\ref{adjoint}, which is essentially due to
Ralph Freese,\index{Freese, R.}  asserts that this functor
has a left adjoint given by the ideal lattice functor.  Here
if $h\colon \m k\to \m l$ is a homomorphism between
lattices, then $\vr I(h)\colon \vr I(\m k)\to \vr I(\m l)$
is defined~by
\[
\vr I(h)(I) = \big\{z\in L\mid
\exists x\in I \big(z\leq h(x)\big)\big\}\,.
\]

\begin{thm}\label{adjoint}
  Restriction to $\m l$ is a natural bijection
  \[
  \mid\colon \vr L_{MC}\big(\vr I(\m l), \m k\big) \to \vr
  L\big(\m l, F(\m k)\big).\]
\end{thm}

(We are identifying $\m l$ with the sublattice of $\vr I(\m
l)$ consisting of principal ideals.)

\begin{proof}
  If $h\colon \vr I(\m l)\to\m k$ is a meet continuous
  lattice homomorphism, then the restriction $h|_{\m
    l}\colon \m l\to F(\m k)$ is evidently a lattice
  homomorphism (since lattices are reducts of meet
  continuous lattices).  If $g\colon \m l\to F(\m k)$ is a
  lattice homomorphism, then it is proved in Lemma~5.1 of
  \cite{freese1} that the function
  \[
  \widehat{g}\colon \vr I(\m l)\to\m k\colon I\mapsto
  \bigvee \{ g(x)\mid x\in I\}
  \]
  is a meet continuous lattice homomorphism whose
  restriction to~$\m l$ is~$g$.  It is the unique extension
  of $g$ to $\vr I(\m l)$, since $\vr I(\m l)$ is generated
  under~$\bigvee$ by~$L$.  This proves that restriction
  to~$\m l$ is a bijection between hom-sets. The naturality
  is left as an exercise. (See Chapter IV of~\cite{maclane}
  for the method.)
\end{proof}

The adjunction from $\vr L$ to $\vr L_{MC}$ may be composed
with the adjunction from $\vr S\vr E\vr T$ to $\vr L$ (given
by the free lattice and forgetful functors) to produce a
left adjoint $\vr S\vr E\vr
T\stackrel{\textit{free}}{\longrightarrow} \vr
L\stackrel{\vr I}{\longrightarrow} \vr L_{MC}$ to the
composite forgetful functor $\vr L_{MC}\to \vr L \to \vr
S\vr E\vr T$. This adjoint establishes the existence and
structure of free meet continuous lattices.

\begin{cor}\label{freese_thm}
  If\/ $\m f = \m f_{\vr L}(X)$ is the free lattice
  generated by~$X$, then $\vr I(\m f)$ is a free meet
  continuous lattice over the set $\{(x]\mid x\in X\}$.
\end{cor}

Since ${\vr L}_{MC}$ is definable by identities and has free
algebras of all ranks, Birkhoff's HSP theorem is valid for
${\vr L}_{MC}$: a subclass of $\vr L_{MC}$ is definable by
identities if and only if it is closed under the formation
of homomorphic images, subalgebras and products.  Hence the
proof of Theorem~\ref{proj+si_implies_splitting} is valid
for varieties of meet continuous lattices. Using known
methods it can be shown that the projective subdirectly
irreducible members of ${\vr L}_{MC}$ are the finite, lower
bounded, subdirectly irreducible lattices satisfying $\W$.
This includes all lattices that are projective and
subdirectly irreducible in ${\vr L}$, along with some others
(such as~$\m d_1$).


\section{Maltsev Conditions}\label{maltsev_conditions}\index{Maltsev condition|(}

A {\bf strong Maltsev 
condition}\index{Maltsev condition!strong --- defined formally} 
is a primitive
positive sentence in the language of clones. This means that
it is a first-order sentence of the form ``$\exists
\bigwedge \left( \mbox{\rm atomic}\right)$'' about clones.
A {\bf Maltsev 
condition}\index{Maltsev condition!defined formally} 
is a countably infinite disjunction
$\bigvee_{i\in {\omega}} \sigma_i$ where the $\sigma_i$ are
strong Maltsev conditions that get weaker as $i$ increases
(i.e., $\sigma_i\vdash \sigma_{i+1}$ for all $i$).
A~variety {\bf 
satisfies}\index{Maltsev condition!satisfaction of}\index{satisfies!a Maltsev condition} 
a (strong) Maltsev condition if its clone does.

The concept of a Maltsev condition can be reformulated in
terms of clone homomorphisms. To each strong Maltsev
condition~$\exists F\bigwedge \Sigma$ corresponds a finite
presentation $\lb F\mid \Sigma\,\rb$ of a clone.  If $\vr U$
is a variety whose clone has this presentation, then $\vr V$
satisfies the strong Maltsev condition~$\exists F\bigwedge
\Sigma$ if and only if~$\vr U\leq \vr V$.  Similarly, to
each Maltsev condition~$\bigvee_{i\in {\omega}} \sigma_i$
corresponds a sequence $\cdots \leq \vr U_2 \leq \vr U_1
\leq \vr U_0$ of~varieties with finitely presented clones. A
variety $\vr V$ satisfies~$\bigvee_{i\in {\omega}} \sigma_i$
if and only if $\vr U_i\leq \vr V$ for some~$i$.

Our practice will be to express Maltsev conditions
informally. For example, we will express the fact that
variety $\vr V$ satisfies the strong Maltsev condition
\begin{equation}\label{cp_cond}
\exists p \left({\rm comp}_2^3(p,\pi_1^2,\pi_2^2,\pi_2^2) \approx \pi_1^2
\;\;\&\;\;
{\rm comp}_2^3(p,\pi_2^2,\pi_2^2,\pi_1^2) \approx \pi_1^2\right),
\end{equation}
which expresses the fact that $\vr V$ has a Maltsev term, by
saying that $\vr V$~has a ternary term $p$ such that the
identities $p(x,y,y)\approx x$ and $p(y,y,x)\approx x$ hold
in~$\vr V$.

An $n$-ary element $f$ of a clone $\vr C$ is {\bf
  idempotent}\index{idempotent!element of a clone} if
\[
{\rm comp}_1^n(f,\pi_1^1,\pi_1^1,\ldots,\pi_1^1) = \pi_1^1\,.
\]
If $\vr C$ is the clone of a variety $\vr V$, this means
that $f(x,x,\ldots,x)\approx x$ is satisfied in $\vr V$.
The idempotent elements of $\vr C$ form a subclone, $\Id(\vr
C)$.\index{$\Id$} The {\bf idempotent
  reduct}\index{idempotent reduct} of $\vr V$ is the variety
$\Id(\vr V):={\rm Var}({\Id(\Clo(\vr V))})$ that is
associated to the idempotent subclone of $\Clo(\vr V)$.  A
(strong) Maltsev condition is {\bf
  idempotent}\index{idempotent!Maltsev condition}\index{Maltsev condition!idempotent} 
if for each
term $f$ in the condition the identity
$f(x,x,\ldots,x)\approx x$ is a consequence of the
identities of the Maltsev condition.

Two (strong) Maltsev conditions are {\bf
  equivalent}\index{Maltsev condition!equivalent}\index{equivalent!Maltsev conditions}
if they define the same class of varieties.

\begin{lm} 
  Any idempotent strong Maltsev condition is equivalent to
  one of the form $\exists F \bigwedge \Sigma$ where $F =
  \{h, k\}$, $h$ is $n$-ary and $k$ is $n^2$-ary, and
  $\Sigma$ consists of the identities
  \begin{enumerate}
  \item[(i)] $h(x,x,\ldots, x)\approx x$,
  \item[(ii)] $k(x_{11}, \ldots, x_{n n}) \approx
    h\big(h(x_{11}, \ldots, x_{1 n}), \ldots, h(x_{n1},
    \ldots, x_{n n})\big)$, plus
  \item[(iii)] finitely many identities of the form
    $k(\mbox{\rm variables})\approx k(\mbox{\rm
      variables})$.
  \end{enumerate}
\end{lm}

\noindent
The proof of this lemma is part of the proof of Lemma~9.4
of~\cite{hobby-mckenzie}.

A strong Maltsev condition $\exists F \bigwedge \Sigma$ is
{\bf linear}\index{Maltsev condition!linear}\index{linear!Maltsev condition} 
if each
identity in~$\Sigma$ has the form $p(\mbox{\rm
  variables})\approx q(\mbox{\rm variables})$ where $p, q\in
F\cup\{\mbox{\rm variables}\}$, while a Maltsev condition
$\bigvee_{i\in {\omega}} \sigma_i$ is linear if each
$\sigma_i$ is.

\begin{lm}\label{id_linear} 
  Any idempotent linear strong Maltsev condition is
  equivalent to one of the form $\exists F \bigwedge \Sigma$
  where $F = \{f\}$, and\/ $\Sigma$ consists of the
  identities
  \begin{enumerate}
  \item[(i)] $f(x,x,\ldots, x)\approx x$, plus
  \item[(ii)] finitely many identities of the form
    $f(\mbox{\rm variables}) \hskip -1pt \approx \hskip -1pt
    f(\mbox{\rm variables})$.
  \end{enumerate}
\end{lm}

\noindent
This is proved by slightly modifying the proof in
\cite{hobby-mckenzie} of the preceding lemma.

All of the Maltsev conditions considered in this monograph
are idempotent and linear, and will usually be expressible
in the form described in Lemma~\ref{id_linear} using only
two variables.  We introduce a notation for such
$2$-variable Maltsev conditions now.

Let $f$ be an $n$-ary operation symbol and let $N =
\{1,\ldots, n\}$.  For each $U\subseteq N$, let $f_U(x,y)$
denote the term obtained from $f(x_1,\ldots, x_n)$ by
substituting $x$ for $x_i$ if $i\in U$ and $y$ for $x_j$ if
$j\notin U$.  Observe that any identity of the form
\[
f(\mbox{\rm variables})\approx f(\mbox{\rm variables})
\]
that uses only the variables $x$ and $y$ may be rewritten in
the form $f_U(x,y)\approx f_V(x,y)$ where $U$ is the set of
positions where $x$ appears on the left of the identity and
$V$ is the set of positions where $x$ appears on the
right. This particular identity may be abbreviated
by~\hbox{$U\equiv V$.}  In order to encode a family $\vr F$
of 2-variable identities of the form $f(\mbox{\rm
  variables})\approx f(\mbox{\rm variables})$, we define
$\vr B(f)$\index{$\vr B(f)$} to be the Boolean algebra of
subsets of $N$ equipped with an equivalence relation $E$
where $U\equiv_{E} V$ holds if and only if $f_U(x,y)\approx
f_V(x,y)$ is a consequence of the identities in~$\vr F$.

\begin{exmp}\label{maltsev_term}
  The statement that $p$ is a Maltsev term for $\vr V$ may
  be expressed by saying that $\vr V$ satisfies identities
  of the form $p(\mbox{\rm variables})\approx p(\mbox{\rm
    variables})$ if the idempotence of $p$ is assumed,
  namely by $p(x,y,y)\approx p(x,x,x)$ and $p(y,y,x)\approx
  p(x,x,x)$.  Here $N = \{1, 2, 3\}$ and ${\mathcal B}(p)$
  is the Boolean algebra of Figure~\ref{fig3}
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1.2mm}
\begin{picture}(30,40)
\put(0,15){\circle*{1.1}}
\put(0,25){\circle*{1.1}}
\put(15,5){\circle*{1.1}}
\put(15,15){\circle*{1.1}}
\put(15,25){\circle*{1.1}}
\put(15,35){\circle*{1.1}}
\put(30,15){\circle*{1.1}}
\put(30,25){\circle*{1.1}}
\put(0,15){\line(3,-2){15}}
\put(0,15){\line(3,2){15}}
\put(0,15){\line(0,1){10}}
\put(15,15){\line(-3,2){15}}
\put(15,15){\line(3,2){15}}
\put(15,15){\line(0,-1){10}}
\put(15,35){\line(-3,-2){15}}
\put(15,35){\line(0,-1){10}}
\put(15,35){\line(3,-2){15}}
\put(30,15){\line(-3,-2){15}}
\put(30,15){\line(-3,2){15}}
\put(30,15){\line(0,1){10}}
\put(-7,14){$\{1\}$}
\put(-10,24){$\{1,2\}$}
\put(14,0){$\emptyset$}
\put(16,12){$\{2\}$}
\put(14,37){$N$}
\put(6,26){$\{1,3\}$}
\put(32,14){$\{3\}$}
\put(32,24){$\{2,3\}$}
\end{picture}
\end{center}
\caption{${\vr B}(p)$}\label{fig3}
\end{figure}
with an equivalence relation $E$ where $N\equiv_E
\{1\}\equiv_E \{3\}$, since \hbox{$p(x,x,x)\approx
  p(x,y,y)\approx p(y,y,x)$} are consequences of the
starting identities, and $\emptyset \equiv_E
\{1,2\}\equiv_E\{2,3\}$, since $p(y,y,y)\approx
p(x,x,y)\approx p(y,x,x)$ are also consequences.
\end{exmp}

A subset of $\vr B(f)$ is 
{\bf closed}\index{$\vr B(f)$!closed filter of}\index{closed filter of $\vr B(f)$}
if it is a union of $E$-classes. The next theorem is due to
Walter Taylor.\index{Taylor, W.}

\begin{thm}\label{wtaylor_term}
  The following are equivalent for a variety $\vr V$.
  \begin{enumerate}
  \item[(1)] $\vr V$ satisfies a nontrivial idempotent
    Maltsev condition (i.e., one that fails in some
    variety).
  \item[(2)] $\vr V$ satisfies an idempotent Maltsev
    condition that fails in the variety of sets.
  \item[(3)] For some $n$, $\vr V$ has an idempotent $n$-ary
    term $f$ such that $\vr B(f)$ has no closed ultrafilter.
  \end{enumerate}
\end{thm}

\begin{proof}
  See Corollary~5.3 of \cite{taylor3}.
\end{proof}

We will make frequent use of Theorem~\ref{wtaylor_term}, so
we introduce the following terminology.

\begin{df}\label{taylor_term}
  A term $f$ is a {\bf Taylor term}\index{Taylor term} for a
  variety $\vr V$ if $\vr V$ satisfies
  $f(x,x,\ldots,x)\approx x$ and enough other identities of
  the form $f_U(x,y)\approx f_V(x,y)$ so that $\vr B(f)$ has
  no closed ultrafilter.
\end{df}

Note that the word `linear' does not appear in
Conditions~(1) and~(2) of~Theorem~\ref{wtaylor_term}
although the Maltsev condition from~(3) is linear.  The fact
that any variety satisfying a nontrivial idempotent Maltsev
condition also satisfies one that is linear is the
nontrivial part of the theorem.

Condition~(3) of Theorem~\ref{wtaylor_term} means that if
$\vr U$ is the ultrafilter of subsets of $N$ containing the
singleton $\{i\}$, then $\vr U$ is not closed. Therefore
there is some $U\in \vr U$ and some $V\notin \vr U$ such
that $\vr V$ satisfies $f_U(x,y)\approx f_V(x,y)$.  Another
way to express Condition~(3) is that $\vr V$~satisfies an
identity of the type $f(\mbox{\rm variables})\approx
f(\mbox{\rm variables})$ for each $1\leq i\leq n$, where all
variables are $x$ or $y$, and $x$ appears in the $i$-th
position on the left while $y$ appears in the $i$-th
position on the right.  An identity of this type will be
called an 
{\bf $i$-th Taylor identity}\index{Taylor identity} 
for~$f$.

A result similar to Taylor's was discovered by David Hobby
and Ralph McKenzie.\index{Hobby, D.}\index{McKenzie, R.}

\begin{thm}\label{hobby-mckenzie_term}
  The following are equivalent for a variety $\vr V$.
  \begin{enumerate}
  \item[(1)] $\vr V$ satisfies an idempotent Maltsev
    condition that fails in the variety of semilattices.
  \item[(2)] For some $n$, $\vr V$ has an idempotent $n$-ary
    term $f$ such that $\vr B(f)$ has no closed, proper,
    nonempty lattice filter.
  \end{enumerate}
\end{thm}

\begin{proof}
  See Lemma~9.5 of \cite{hobby-mckenzie}.
\end{proof}

\begin{df}\label{hm_term}
  A term $f$ is a {\bf Hobby--McKenzie
    term}\index{Hobby--McKenzie term} for a variety~$\vr V$
  if $\vr V$ satisfies $f(x,x,\ldots,x)\approx x$ and enough
  other identities of the form $f_U(x,y)\approx f_V(x,y)$ so
  that $\vr B(f)$ has no closed, proper, nonempty lattice
  filter.
\end{df}

The term $p$ from Example~\ref{maltsev_term} is a
Hobby--McKenzie term, and therefore also a Taylor term. To
see this, suppose that $\vr F$ is a closed, proper, nonempty
lattice filter of $\vr B(p)$. Then since $\vr F$ is a
nonempty filter it contains the top element $N$. Since $\vr
F$ is closed it must contain $\{1\}\; ( \equiv_E N)$ and
also $\{3\}\; ( \equiv_E N)$. Since $\vr F$ is a lattice
filter, it contains $\{1\}\cap \{3\} = \emptyset$.  But
any lattice filter containing $\emptyset$ is
improper.\index{Maltsev condition|)}

\section{The Term Condition}\label{the_term_condition}\index{term condition|(}

Let $\m a = \lb A; F\rb$ be an algebra.  An $n$-ary relation
$R\subseteq A^n$ is 
{\bf compatible}\index{compatible relation} 
if it is a subalgebra of $\m a^n$.  If $\m b$ is
a subalgebra of $\m a$, then the {\bf restriction} of a
relation $R\subseteq A^n$ to $\m b$ is $R|_{\m b} := R\cap
B^n$.  If $\delta$~is a congruence on~$\m a$, then $R/\delta
:= \{ (a_1/\delta, \ldots, a_n/\delta) \mid (a_1,\ldots,
a_n)\in R\}$. Both $R|_{\m b}$ and $R/\delta$ are compatible
if $R$ is.  If $\delta$ is a congruence, then $R$~is {\bf
  $\delta$-closed}\index{$\delta$-closed} if $R =
\delta\circ R\circ\delta$. (I.e., if
$\wec{a}\wrel{\delta}\wec{b}\wrel{R}\wec{c}\wrel{\delta}\wec{d}$
implies $\wec{a}\wrel{R}\wec{d}$.)

A compatible, reflexive, symmetric binary relation is called
a {\bf tolerance}.\index{tolerance} We will usually denote
tolerances by upper case italic letters: $R$, $S$, $T,
\ldots$.  A compatible equivalence relation (i.e., a
transitive tolerance) is a {\bf
  congruence},\index{congruence!defined formally} and
congruences will usually be denoted by lower case Greek
letters: $\alpha$, $\beta$, $\gamma, \ldots$.  The tolerance
or congruence generated by a set $X\subseteq A\times A$ is
usually denoted by $\Tg a (X)$\index{$\Tg a (X)$} or $\Cg a
(X)$\index{$\Cg a (X)$} respectively, although if $X$
contains only a few pairs then we may write, for example,
$\Cg a (a,b)$ or $\Cg a\big((a,b), (c,d)\big)$ instead.  If
$T$ is a tolerance on~$\m a$, then a maximal subset
$B\subseteq A$ such that $B\times B\subseteq T$ is called a
{\bf block}\index{block of a tolerance}\index{tolerance!block of} 
of $T$. If $T$ is in
fact a congruence, then a block is the same thing as a
congruence class. A tolerance or congruence is {\bf
  trivial}\index{trivial!congruence or tolerance}\index{congruence!trivial}\index{tolerance!trivial}
if it is the equality relation and {\bf nontrivial}
otherwise.

The collection of congruences on $\m a$, ordered by
inclusion, is an algebraic lattice which is
denoted\index{Con(A)@$\Conb(\m a)$} $\Conb(\m a)$. Its least
and largest elements are denoted $0$ and $1$.  Meet and join
are denoted $\wedge$ and $\vee$ and are computed by
$\alpha\wedge\beta = \alpha\cap\beta$ and $\alpha\vee\beta
= \mbox{\rm tr.cl.}(\alpha\cup\beta)$ where $\mbox{\rm
  tr.cl.}$ represents transitive closure.

An $m$-ary {\bf polynomial 
operation}\index{polynomial operation} 
of $\m a$ is an operation $f\colon A^m\to A$
such that $f(x_1,\ldots,x_m) = t^{\m a}(x_1,\ldots,x_m,
\wec{a})$ for some $(m+n)$-ary term~$t$ and some tuple
$\wec{a}\in A^n$.

If $S$ and $T$ are tolerances on $\m a$, then an {\bf $S,
  T$-matrix}\index{$S, T$-matrix}\index{S,T@$S, T$-matrix}
is a $2\times 2$ matrix of elements of $\m a$ of the form
\[
\left[\begin{array}{cc}
p&q\\
r&s
\end{array}\right]=
\left[\begin{array}{cc}
f(\wec{a},\wec{u})&f(\wec{a},\wec{v})\\
f(\wec{b},\wec{u})&f(\wec{b},\wec{v})
\end{array}\right]\]
where $f(\wec{x},\wec{y})$ is an $(m+n)$-ary polynomial of
$\m a$, $\wec{a} \wrel{S} \wec{b}$, and
$\wec{u}\wrel{T}\wec{v}$.  The set of all $S, T$-matrices is
denoted $M(S, T)$.\index{$M(S, T)$}

Since tolerances are compatible with all polynomial
operations, any two elements in the same row of an $S,
T$-matrix are $T$-related and any two elements in the same
column are $S$-related.

The fact that $S$ and $T$ are symmetric relations implies
that $M(S,T)$ is closed under interchanging rows or columns:
\[
\left[\begin{array}{cc}
p&q\\
r&s
\end{array}\right]
\in M(S,T)\Leftrightarrow
\left[\begin{array}{cc}
r&s\\
p&q
\end{array}\right]
\in M(S,T)\Leftrightarrow
\left[\begin{array}{cc}
s&r\\
q&p
\end{array}\right]
\in M(S,T)\,.
\]
If $S=T$, then $M(S,T)=M(T,T)$ is also closed under
transpose, as one sees by interchanging the roles of
$\wec{x}$ and $\wec{y}$ in the
polynomial~$f(\wec{x},\wec{y})$ that defines a given matrix.

\begin{df}\label{centralizer_def}
  Let $S$ and $T$ be tolerances on an algebra $\m a$, and
  let $\delta$ be a congruence on $\m a$.  If
  $p\equiv_{\delta} q$ implies that $r\equiv_{\delta} s$
  whenever
  \begin{equation}\label{S,T-matrix}
    \left[\begin{array}{cc}
        p&q\\
        r&s
      \end{array}\right]\in M(S, T),
  \end{equation}
  then we say that {\bf $\C C(S,T;\delta)$ 
    holds},\index{$\C C(S,T;\delta)$} 
  or {\bf $S$ centralizes $T$ modulo
    $\delta$}.\index{centralizes}
\end{df}

By interchanging the rows of matrices one sees that $\C
C(S,T;\delta)$ holds if and only if
\[
p\equiv_{\delta}q\quad\miff\quad r\equiv_{\delta} s
\]
for every $S,T$-matrix in (\ref{S,T-matrix}).

The {\bf $S, T$-term
  condition}\index{term condition!defined} is the condition
$\C C(S, T; 0)$. There are other similar conditions called
term conditions that we will meet later, but this is the
original one.

When establishing that the implication defining $\C
C(S,T;\delta)$ holds, or when making use of the fact, we may
use underlining to highlight places in equations or
expressions where changes are to be made. For example, we
may write the implication defining $\C C(S,T;\delta)$ in the
following form: If
\[
f(\underline{\wec{a}}, \wec{u}) \equiv_{\delta} 
f(\underline{\wec{a}}, \wec{v})\,,
\]
then 
\[
f(\underline{\wec{b}}, \wec{u}) \equiv_{\delta} 
f(\underline{\wec{b}}, \wec{v})\,.
\]

The relation $\C
C({\hphantom{S}},{\hphantom{T}};{\hphantom{T}})$ is called
the {\bf centralizer relation}.\index{centralizer relation}
The reason that this terminology is used is that when $\m a$
is a group and $S, T$ and $\delta$ are congruences on $\m
a$, then $\C C(S, T; \delta)$ holds if and only if $[S,
T]\leq \delta$ (see Chapter~1 of \cite{freese-mckenzie}).

The basic properties of the centralizer relation are
enumerated in the following theorem.

\begin{thm}\label{basic_centrality}
  Let $\m a$ be an algebra with tolerances $S$, $S'$, $T$,
  $T'$ and congruences $\alpha$, $\alpha_i$, $\beta$,
  $\delta$, $\delta'$, $\delta_j$.  The following are true.
  \begin{enumerate}
  \item[(1)] {\rm (Monotonicity in the first two variables)}
    If\/ $\C C(S,T;\delta)$ holds and $S'\subseteq S$,
    $T'\subseteq T$, then $\C C(S',T';\delta)$ holds.
  \item [(2)] $\C C(S,T;\delta)$ holds if and only if\/ $\C
    C(\Cg a(S),T;\delta)$ holds.
  \item [(3)] $\C C(S, T; \delta)$ holds if and only if\/
    $\C C(S, \delta\circ T\circ \delta; \delta)$ holds.
  \item [(4)] If\/ $T\cap \delta = T\cap\delta'$, then
    $\C C(S,T;\delta)\miff \C C(S,T;\delta')$.
  \item [(5)] {\rm (Semidistributivity in the first
      variable)} If\/ $\C C(\alpha_i,T;\delta)$ holds for
    all $i\in I$, then\/ $\C C(\bigvee_{i\in
      I}\alpha_i,T;\delta)$ holds.
  \item[(6)] If\/ $\C C(S,T;\delta_j)$ holds for all $j\in
    J$, then\/ $\C C(S,T;\bigwedge_{j\in J}\delta_j)$ holds.
  \item [(7)] If\/ $T\cap\big(S\circ (T\cap\delta)\circ
    S\big) \subseteq\delta$, then\/ $\C C(S,T;\delta)$
    holds.
  \item [(8)] If\/
    $\beta\wedge\big(\alpha\vee(\beta\wedge\delta)\big)
    \leq\delta$, then\/ $\C C(\alpha,\beta;\delta)$ holds.
  \item [(9)] Let\/ $\m b$ be a subalgebra of\/ $\m a$.
    If\/ $\C C(S, T; \delta)$ holds in $\m a$, then\/ $\C
    C(S|_{\m b}, T|_{\m b}; \delta|_{\m b})$ holds in $\m
    b$.
  \item [(10)] If\/ $\delta'\leq \delta$, then the
    relation\/ $\C C(S, T; \delta)$ holds in $\m a$ if and
    only if\/ $\C C(S/\delta', T/\delta'; \delta/\delta')$
    holds in $\m a/\delta'$.
  \end{enumerate}
\end{thm}

\begin{proof}
  Item (1) follows from the fact that $M(S',T')\subseteq
  M(S,T)$.

  For~(2), $\C C\big(\Cg a(S),T;\delta\big) \mimplies \C
  C(S,T;\delta)$ follows from (1), since $S\subseteq \Cg
  a(S)$.  For the reverse implication (and also for the
  proof of item~(5)), we will argue that if $S_i$ is a
  tolerance, $\C C(S_i, T;\delta)$ holds for all $i\in I$,
  and $\alpha := {\rm tr.cl.}\left(\bigcup_{i\in I}
    S_i\right)$, then $\C C(\alpha, T; \delta)$. (To
  complete the proof of (2) we need this only when $|I| =
  1$, while in (5) we need it only when the $S_i$ are
  congruences.)

  Choose any matrix in $M(\alpha, T)$. If it is
  \[
  \left[\begin{array}{cc}
      p&q\\
      r&s
    \end{array}\right] = 
  \left[\begin{array}{cc}
      f(\wec{a},\wec{u})&f(\wec{a},\wec{v})\\
      f(\wec{b},\wec{u})&f(\wec{b},\wec{v})
    \end{array}\right],
  \]
  then $\wec{a}$ is related to $\wec{b}$ by ${\rm
    tr.cl.}\left(\bigcup_{i\in I} S_i\right)$.  It is easy
  to see that there exist tuples $\wec{a} =
  \wec{a}_0\wrel{S_{i_1}} \wec{a}_1\wrel{S_{i_2}} \cdots
  \wrel{S_{i_n}} \wec{a}_{n} = \wec{b}$. These tuples
  determine matrices
  \[
  \left[\begin{array}{cc}
      p_k&q_k\\
      p_{k+1}&q_{k+1}
    \end{array}\right] :=
  \left[\begin{array}{cc}
      f(\wec{a}_k,\wec{u})&f(\wec{a}_k,\wec{v})\\
      f(\wec{a}_{k+1},\wec{u})&f(\wec{a}_{k+1},\wec{v})
    \end{array}\right]
  \in M(S_{i_{k+1}}, T)\,.
  \]
  We must show that $p\equiv_{\delta} q$ implies
  $r\equiv_{\delta} s$, so assume that $p\equiv_{\delta}
  q$. This is the same as $p_0\equiv_{\delta} q_0$, and so
  by induction (using that $\C C(S_{i_k}, T;\delta)$ holds
  for all $k$) we get that $p_{k}\equiv_{\delta} q_{k}$ for
  all $k$.  Therefore $r = p_n \equiv_{\delta} q_n = s$.
  This completes the proofs of (2) and (5).

  For (3), the implication $\C C(S, \delta\circ T\circ
  \delta; \delta) \mimplies \C C(S, T; \delta)$ follows
  from~(1), since $T\subseteq \delta\circ T\circ \delta$.
  For the reverse implication, assume that $\C C(S, T;
  \delta)$ holds, that
  \[
  \left[\begin{array}{cc}
      p&q\\
      r&s
    \end{array}\right] = 
  \left[\begin{array}{cc}
      f(\wec{a},\wec{u})&f(\wec{a},\wec{v})\\
      f(\wec{b},\wec{u})&f(\wec{b},\wec{v})
    \end{array}\right]\in M(S, \delta\circ T \circ \delta)\,, 
  \]
  and that $p\equiv_{\delta} q$.  There exist tuples
  $\wec{u'}$ and $\wec{v'}$ such that $\wec{u}\wrel{\delta}
  \wec{u'} \wrel{T}\wec{v'}\wrel{\delta}\wec{v}$. The matrix
  \[
  \left[\begin{array}{cc}
      p'&q'\\
      r'&s'
    \end{array}\right] = 
  \left[\begin{array}{cc}
      f(\wec{a},\wec{u'})&f(\wec{a},\wec{v'})\\
      f(\wec{b},\wec{u'})&f(\wec{b},\wec{v'})
    \end{array}\right]
  \]
  is an $S, T$-matrix. Moreover,
  \[
  p' = f(\wec{a},\wec{u'}) \wrel{\delta} f(\wec{a},\wec{u})
  = p \wrel{\delta} q = f(\wec{a},\wec{v}) \wrel{\delta}
  f(\wec{a},\wec{v'}) = q'\,.
  \]
  Since $\C C(S, T; \delta)$ holds, it follows that
  $r'\equiv_{\delta} s'$. Hence
  \[
  r = f(\wec{b},\wec{u}) \wrel{\delta} f(\wec{b},\wec{u'}) =
  r' \wrel{\delta} s' = f(\wec{b},\wec{v'}) \wrel{\delta}
  f(\wec{b},\wec{v}) = s\,,
  \]
  or $r\equiv_{\delta} s$.  This establishes $\C C(S,
  \delta\circ T\circ\delta; \delta)$.

  For (4), recall that elements in the same row of an $S,
  T$-matrix are $T$-related. So if $\left[\begin{array}{cc}
      p & q\\ r& s\end{array}\right]\in M(S, T)$, then since
  $T\cap \delta = T\cap \delta'$ we get that
  \[
  p\equiv_{\delta} q \wrel{\miff} p\equiv_{T\cap \delta} q
  \wrel{\miff} p\equiv_{T\cap \delta'} q \wrel{\miff}
  p\equiv_{\delta'} q\,,
  \]
  and
  \[
  r\equiv_{\delta} s \wrel{\miff} r\equiv_{T\cap \delta} s
  \wrel{\miff} r\equiv_{T\cap \delta'} s \wrel{\miff}
  r\equiv_{\delta'} s\,.
  \]
  Therefore the implication $p\equiv_{\delta} q \mimplies
  r\equiv_{\delta} s$ is equivalent to the implication
  $p\equiv_{\delta'} q \mimplies r\equiv_{\delta'} s$.

  For (6), assume that $\left[\begin{array}{cc} p & q\\ r&
      s\end{array}\right]\in M(S, T)$.  If $p\equiv q
  \pmod{\bigwedge \delta_j}$, then $p\equiv q
  \pmod{\delta_j}$ for all $j$. Since $\C C(S, T; \delta_j)$
  holds for all $j$ we get that $r\equiv s \pmod{\delta_j}$
  for all $j$, or equivalently that $r\equiv s
  \pmod{\bigwedge \delta_j}$.  This shows that $\C C(S, T;
  \bigwedge_{j\in J} \delta_j)$ holds.

  For (7), choose an $S, T$-matrix $M =
  \left[\begin{array}{cc} p & q\\ r& s\end{array}\right]$.
  Assume that $p\equiv_{\delta} q$. Since the elements in
  the same row of $M$ are $T$-related and the elements in
  the same column are $S$-related, we have $r\wrel{S} p
  \wrel{T\cap\delta} q\wrel{S} s$. Moreover, $r\wrel{T} s$
  since these elements belong to the same row.  Together
  this yields that $r\wrel{T\cap \big(S\circ
    (T\cap\delta)\circ S\big)} s$.  By the assumption in
  (7), this implies that $r\equiv_{\delta} s$.  This
  proves~(7).

  For item (8), if
  $\beta\wedge\big(\alpha\vee(\beta\wedge\delta)\big)
  \leq\delta$, then $\beta\cap\big(\alpha\circ
  (\beta\cap\delta)\circ \alpha\big)\leq\delta$, so $\C
  C(\alpha,\beta;\delta)$ holds by (7).

  Item (9) holds because any instance of the implication in
  Definition~\ref{centralizer_def} defining $\C C(S|_{\m b},
  T|_{\m b}; \delta|_{\m b})$ in $\m b$ is an instance of
  the implication defining $\C C(S, T; \delta)$ in $\m a$.

  For item (10), it suffices to observe that, when
  $\delta'\leq \delta$,
  \[
  \left[
    \begin{array}{cc}
      p'/\delta' & q'/\delta'\\ 
      r'/\delta' & s'/\delta'
    \end{array}\right]\in M(S/\delta', T/\delta')
  \]
  if and only if there exist $p\equiv_{\delta'} p'$,
  $q\equiv_{\delta'} q'$, $r\equiv_{\delta'} r'$, and
  $s\equiv_{\delta'} s'$ with
  \[
  \left[
    \begin{array}{cc}
      p & q\\ 
      r & s
    \end{array}\right]\in M(S, T), 
  \]
  and that $p\equiv_{\delta} q\Leftrightarrow
  p'/\delta'\equiv_{\delta/\delta'} q'/\delta'$ and
  $r\equiv_{\delta} s\Leftrightarrow
  r'/\delta'\equiv_{\delta/\delta'} s'/\delta'$.
\end{proof}

\begin{df}\label{abelian_df}
  The {\bf commutator}\index{commutator} of $S$ and $T$,
  denoted by~$[S,T]$, is the least congruence $\delta$ such
  that $\C C(S,T;\delta)$ holds.  $T$ is {\bf
    abelian}\index{abelian!tolerance}\index{tolerance!abelian}
  if $[T,T] = 0$.  An algebra $\m a$ is {\bf
    abelian}\index{abelian!algebra}\index{algebra!abelian}
  if its largest congruence is.
\end{df}

By Theorem ~\ref{basic_centrality}~(6), the class of all
$\delta$ such that $\C C(S, T ; \delta)$ holds is closed
under complete meet, so there is a least such $\delta$. This
implies that $[S,T]$ exists for any two tolerances $S$ and
$T$.

It is a well known fact, easily derivable from the
definitions, that $\m a$~is abelian if and only if the
diagonal of $A\times A$ is a class of a congruence of $\m
a\times \m a$.

\begin{df}\label{centralizers}
  The {\bf centralizer of $T$ modulo $\delta$}, denoted by
  $(\delta:T)$,\index{$(\delta:T)$} is the largest
  congruence $\alpha$ on $\m a$ such that $\C C(\alpha, T ;
  \delta)$ holds.
\end{df}

By Theorem ~\ref{basic_centrality}~(5), the class of all
$\alpha$ such that $\C C(\alpha, T ; \delta)$ holds is
closed under complete join, so there is a largest such
$\alpha$. This implies that $(\delta:T)$ exists for every
$\delta$ and $T$.  By Theorem ~\ref{basic_centrality}~(2),
the centralizer $(\delta: T)$ contains every tolerance $S$
such that $\C C(S, T; \delta)$ holds.\index{term condition|)}



\section{Congruence Identities}\label{congruence_identities}

If $\vr V$ is a variety of algebras, then any lattice
identity that holds in the class $\{\Conb(\m a)\mid \m a\in
\vr V\}$ of congruence lattices of algebras in~$\vr V$ is
called a {\bf congruence 
identity}\index{identity!congruence ---}\index{congruence identity!of a variety} 
of $\vr V$. The 
{\bf congruence variety}\index{congruence variety}\index{variety!congruence ---} 
of~$\vr V$, denoted
$\CON(\vr V)$, \index{CON(V)@$\CON(\vr V)$} is the
subvariety of $\vr L$ generated by \hbox{$\{\Conb(\m
  a)\mid\m a\in \vr V\}$,} or alternatively is the variety
of lattices axiomatized by the congruence identities that
hold in $\vr V$.  Similarly, a lattice quasi-identity that
holds in congruence lattices of members of $\vr V$ is a {\bf
  congruence 
quasi-identity}\index{quasi-identity!congruence ---}\index{congruence quasi-identity} 
of $\vr V$.

The following theorem will be used in several places in this
monograph.

\begin{thm}[Cf.\ \cite{czedli-day}]\label{czedli-day-W}
  Let $Q$ be a quasi-identity satisfying~$\W$.  The class of
  varieties satisfying $Q$ as a congruence quasi-identity is
  definable by a set of idempotent Maltsev conditions.
\end{thm}

Since congruence lattices are algebraic, and therefore meet
continuous, it is reasonable to discuss {\bf meet continuous
  congruence 
identities},\index{congruence identity!meet continuous ---} 
which we define to be meet continuous
lattice identities considered as congruence identities.  A
meet continuous congruence identity is {\bf
  trivial}\index{trivial!meet continuous congruence identity} 
if it holds in the congruence lattice of every
algebra and {\bf nontrivial} otherwise.  It follows from
Corollary~\ref{freese_thm} that free meet continuous
lattices are algebraic, so they are isomorphic to congruence
lattices according to the Gr\"{a}tzer--Schmidt
Theorem\index{Gr\"{a}tzer--Schmidt Theorem}
\cite{gratzer-schmidt}.  Therefore a meet continuous lattice
identity is nontrivial as a congruence identity if and only
if it is nontrivial as a meet continuous lattice
identity. Both statements mean the identity fails in some
algebraic lattice.

It is proved in \cite{czedli2} that the class of varieties
that satisfy the meet semidistributive law as a congruence
quasi-identity is definable by a set of idempotent Maltsev
conditions.  An examination of that argument reveals that
what is really proved is that, modulo the axioms defining
${\vr L}_{MC}$, the meet semidistributive law is equivalent
to a meet continuous lattice identity.\footnote{In fact,
  what is \emph{really} shown is that the partial lattice
  $\m p(\sdm)$ in the $\sdm$-configuration is projective
  relative to ${\vr L}_{MC}$.}  In the next theorem, we
prove that many different quasi-identities, including
$\sdm$, are equivalent to meet continuous identities.

\begin{thm}\label{strange-thm}
  Let $p$, $q$, $r$, $s$, $x_1,\ldots,x_n$ be lattice
  variables.  Define $t := p\wedge (q\vee r)$.  Let
  $w_i(p,q,r)$, $1\leq i\leq m$, be ternary lattice words
  such that $w_i(p,q,r)\leq q$ for some $i$, $w_j(p,q,r)\leq
  r$ for some $j$, and $w_k(p,q,r)\leq t$ for all $k$ in the
  free lattice $\m f_{\vr L}(\{p,q,r\})$.  For each~$i$ let
  $x_i^* = (x_i\vee s)\wedge t$.  For arbitrary $n$-ary
  lattice words $u$ and $v$, the quasi-identity
  \begin{equation}\label{strange-quasi-identity}
    \bigwedge_{i=1}^m \big(w_i(p,q,r)\approx s\big)\to 
    u(x_1^*,\ldots,x_n^*)\approx v(x_1^*,\ldots,x_n^*)
  \end{equation}
  is equivalent to an identity in ${\vr L}_{MC}$.
\end{thm}

\begin{proof}
  Let $f_i(x)$ denote the polynomial $w_i(p,q\vee x,r\vee
  x)$ of the free meet continuous lattice $\m f_0 = \m
  f_{{\vr L}_{MC}}(\{p,q,r\})$ (where $1\le i\le m$).  If
  $z\leq t=p\wedge(q\vee r)$, then the idempotence of
  lattice words implies that for any $i$ between $1$ and $m$
  \begin{equation}\label{pol_calculations}
    \begin{array}{r@{}l}
      \underline{z}&{} 
      =w_i(z,z,z)\leq w_i(p,q\vee z,r\vee z)=
      \underline{f_i(z)}\\
      &{} \leq w_i(p,q\vee t,r\vee t)
      \leq w_i(p,q\vee r,q\vee r)\\
      &{} \leq p\wedge\big((q\vee r)\vee (q\vee r)\big) = 
      \underline{t}\,.
    \end{array}
  \end{equation}
  The underlined elements indicate why the principal ideal
  generated by $t=p\wedge(q\vee r)$ is closed under each
  $f_i$, and each $f_i$ is an increasing function on this
  ideal.  It follows from this that the set of all elements
  of the form $f_{i_1}\circ\cdots \circ f_{i_\ell}(0)$,
  $i_j\in \{1,\ldots,m\}$, is an updirected set of elements
  below~$t$.  The supremum of this set,
  $\delta^{\wec{w}}(p,q,r)\in F_0$,
  $\wec{w}:=(w_1,\ldots,w_m)$, is therefore the least common
  fixed point in $\m f_0$ of the polynomials~$f_i$.  Since
  $t=p\wedge(q\vee r)$ is a common fixed point of the $f_i$
  it follows that $\delta^{\wec{w}}\leq t$ in $\m f_0$. Now,
  if $\varphi\colon \m f_0\to \m l$ is a surjective
  homomorphism of meet continuous lattices for which
  $a=\varphi(p)$, $b=\varphi(q)$ and $c=\varphi(r)$, then
  $\varphi\big(\delta^{\wec{w}}(p,q,r)\big) =
  \delta^{\wec{w}}(a,b,c)$ is the supremum of all $f^{\m
    l}_{i_1}\circ\cdots \circ f^{\m l}_{i_k}(0)$, where
  $f^{\m l}_i(x) := w_i^{\m l}(a,b\vee x, c\vee x)$.  By
  an argument like the one in~(\ref{pol_calculations}), each
  $f_i^{\m l}$ is increasing on the principal ideal
  generated by $t^{\m l}(a,b,c)=a\wedge (b\vee c)$ and maps
  this ideal into itself.  Hence $\delta^{\wec{w}}(a,b,c)$
  is the least common fixed point of the polynomials
  $f_i^{\m l}$ in $\m l$.

  For each variable $x_i$, set $x_i^{+}:= \big(x_i\vee
  \delta^{\wec{w}}(p,q,r)\big)\wedge t$, which is an element
  of $\m f:=\m f_{\vr L_{MC}}(\{p,q,r,s,x_1,\ldots,x_n\})$.
  We are now prepared to argue that quasi-identity
  (\ref{strange-quasi-identity}) is equivalent to the
  identity
  \begin{equation}\label{strange-identity}
    u(x_1^{+},\ldots,x_n^{+})\approx v(x_1^{+},\ldots,x_n^{+})
  \end{equation}
  in ${\vr L}_{MC}$.

  Assume first that $\m l$ is a meet continuous lattice
  satisfying (\ref{strange-identity}).  To show that $\m l$
  satisfies (\ref{strange-quasi-identity}), choose any
  assignment of the variables $\{p,q,r,s,\wec{x}\}\to L$
  that satisfies the premises of quasi-identity
  (\ref{strange-quasi-identity}), and let $\psi\colon \m
  f\to\m l$ be the extension of this assignment to a
  homomorphism.  Equivalently, choose a homomorphism $\psi$
  so that if
  \[
  \big(\psi(p),\psi(q),\psi(r),\psi(s),\psi(\wec{x})\big)=
  (a,b,c,d,\wec{g})\,,
  \]
  then $w^{\m l}_i(a,b,c)=d$ for all $i$.  Define
  $e:=\psi(t)=a\wedge (b\vee c)$ and
  $g_i^*:=\psi(x_i^*)=(g_i\vee d)\wedge e$.  We must show
  that $u^{\m l}(\wec{g}^*)=v^{\m l}(\wec{g}^*)$.

  Since $w_i(p,q,r)\leq q$ for some $i$, $w_j(p,q,r)\leq r$
  for some $j$, and $w_k(p,q,r)\leq \big(p\wedge(q\vee
  r)\big)\leq p$ for all $k$ in $\m f_0$, it follows that
  \[
  d=w_i(a,b,c)=w_j(a,b,c)=w_k(a,b,c)\leq a\wedge b\wedge c
  \]
  in $\m l$.  Since $d=w_1(a,b,c)\geq a\wedge b\wedge c$ we
  even have $d = a\wedge b\wedge c$.  Therefore $f_i^{\m
    l}(d)=w^{\m l}_i(a,b\vee d,c\vee d) = w^{\m
    l}_i(a,b,c) = d$ for all $i$, yielding that $d$ is a
  common fixed point of the $f_i$ in~$\m l$.  Since the meet
  continuous sublattice $\m l_0\leq \m l$ that is generated
  by $a$, $b$ and $c$ contains $d$ and has
  $\delta^{\wec{w}}(a,b,c)$ as its least common fixed point
  of the $f_i$'s, it follows that
  $\delta^{\wec{w}}(a,b,c)\leq d$.  Since $d = a\wedge b\wedge
  c\leq f_1(0)\leq \delta^{\wec{w}}(a,b,c)$, we even have $d
  = \delta^{\wec{w}}(a,b,c)$.  This implies that
  $\psi(x_i^{+}) = \psi(x_i^*)=g_i^*$. This gives us the
  desired result
  \[
  u^{\m l}(\wec{g}^*) = \psi\big(u(\wec{x}^+)\big)=
  \psi\big(v(\wec{x}^+)\big)= v^{\m l}(\wec{g}^*),
  \]
  where the middle equality is from
  (\ref{strange-identity}).


  Now suppose conversely that $\m l$ satisfies
  (\ref{strange-quasi-identity}).  To verify that $\m
  l$~satisfies~(\ref{strange-identity}), choose any variable
  assignment in~$\m l$ and extend it to a homomorphism
  $\psi\colon \m f\to \m l$.  We must show that
  $\psi\big(u(\wec{x}^+)\big)= \psi\big(v(\wec{x}^+)\big)$.
  Since the variable $s$ does not appear
  in~(\ref{strange-identity}) we need only consider
  homomorphisms $\psi$ for which
  $\psi(s)=\psi\big(\delta^{\wec{w}}(p,q,r)\big)$, i.e.,
  those of the form
  \[
  \big(\psi(p),\psi(q),\psi(r),\psi(s),\psi(\wec{x})\big)=
  (a,b,c,\delta^{\wec{w}}(a,b,c),\wec{g}).
  \]
  Under such homomorphisms
  \[
  \psi(x_i^+)=\psi(x_i^*) =\big(g_i\vee
  \delta^{\wec{w}}(a,b,c)\big) \wedge \big(a\wedge(b\vee
  c)\big)\,.
  \]

  The element $\delta:=\delta^{\wec{w}}(a,b,c)$ is the least
  common fixed point of the polynomials $f_i^{\m
    l}(x):=w^{\m l}(a,b\vee x,c\vee x)$ in the meet
  continuous sublattice of $\m l$ generated by $a$, $b$ and
  $c$. Thus, if $b^{\delta}:=b\vee \delta$ and
  $c^{\delta}:=c\vee \delta$, we get that
  \[
  \delta = f_i^{\m l}(\delta) =
  w_i(a,b\vee\delta,c\vee\delta) =
  w_i(a,b^{\delta},c^{\delta})
  \]
  for all $i$.  Thus, each premise
  $w_i(a,b^{\delta},c^{\delta})=\delta$ of quasi-identity
  (\ref{strange-quasi-identity}) is satisfied by the
  variable assignment $ \psi^{\delta}\colon
  (p,q,r,s,\wec{x})\mapsto (a,b^{\delta},c^{\delta}, \delta,
  \wec{g}).  $ Since $\m l$ satisfies
  (\ref{strange-quasi-identity}), we get that
  \begin{equation}\label{uv_eqn}
    u^{\m l}\big(\psi^{\delta}(\wec{x}^*)\big) = 
    v^{\m l}\big(\psi^{\delta}(\wec{x}^*)\big)
  \end{equation}
  holds in $\m l$, where
  \[
  \psi^{\delta}(x_i^*) = (g_i\vee \delta)\wedge
  \big(a\wedge(b^{\delta}\vee c^{\delta})\big)\;.
  \]
  But $\delta^{\wec{w}}(p,q,r)\leq p\wedge (q\vee r)$ in $\m
  f_{\vr L_{MC}}(\{p,q,r\})$, so
  \[
  \delta = \delta^{\wec{w}}(a,b,c)\leq a\wedge (b\vee c)
  \]
  in $\m l$. This yields the last equality in
  \[
  a\wedge(b\vee c)\leq a\wedge(b^{\delta}\vee c^{\delta}) =
  a\wedge(b\vee c \vee {\delta}) = a\wedge(b\vee c)\,.
  \]
  This shows that $a\wedge(b^{\delta}\vee c^{\delta}) =
  a\wedge(b\vee c)$, so
  \[
  \psi^{\delta}(x_i^*) = (g_i\vee \delta)\wedge
  \big(a\wedge(b\vee c)\big) = \psi(x_i^+)\,.
  \]
  From this and (\ref{uv_eqn}) we derive that
  \[
  \psi\big(u(\wec{x}^+)\big) = u^{\m
    l}\big(\psi^{\delta}(\wec{x}^*)\big) = v^{\m
    l}\big(\psi^{\delta}(\wec{x}^*)\big) =
  \psi\big(v(\wec{x}^+)\big)\,,
  \]
  as desired.  This proves that (\ref{strange-identity})
  holds in~$\m l$.
\end{proof}

\begin{rem}\label{sdm_rem}
  If $\m l$ is a meet continuous lattice, then for every
  homomorphism $\psi\colon \m f\to \m l$ for which
  $(p,q,r,s,\wec{x})\mapsto (a,b,c,d,\wec{g})$ and
  $w_i(a,b,c)=d$ for all $i$, it must be that $d$ is below
  $\psi(t)=a\wedge (b\vee c)=:e$, since $w_i(p,q,r)\leq t$.
  Call $I:=I[d,e]$ a ``bad interval'' in $\m l$ if it arises
  in this way from elements satisfying the premises of
  quasi-identity~(\ref{strange-quasi-identity}).  For
  each~$i$ we have $\psi(x_i^*) = (g_i\vee d)\wedge e\in I$,
  and conversely for any $h\in I$ we have $h=\psi(x_i^*)$ if
  $\psi$ assigns $h$ to $x_i$. Thus, $\psi(x_i^*)$ may be
  viewed as a typical element of $I$.  Altogether this means
  that the premises of~(\ref{strange-quasi-identity})
  identify which intervals are bad, and the quasi-identity
  itself asserts that all bad intervals satisfy the identity
  $u\approx v$.

  If we choose $m=n=2$, $w_1(p,q,r)=p\wedge q$,
  $w_2(p,q,r)=p\wedge r$, then the bad intervals are the
  $\sdm$-failures.  If $u(x_1,x_2)=x_1$ and
  $v(x_1,x_2)=x_2$, then (\ref{strange-quasi-identity})
  asserts precisely that all $\sdm$-failures satisfy
  $x_1\approx x_2$, i.e., they are trivial.  These choices
  for $w_i$, $u$ and $v$ show that the meet semidistributive
  law is equivalent to a quasi-identity of
  type~(\ref{strange-quasi-identity}), hence to a meet
  continuous identity.

  Consider choosing the premises of
  (\ref{strange-quasi-identity}) in the same way as in the
  previous paragraph, but allowing $u\approx v$ to be some
  other lattice identity.  We obtain from
  Theorem~\ref{strange-thm} that the class of meet
  continuous lattices whose $\sdm$-failures satisfy
  $u\approx v$ is a subvariety of ${\vr L}_{MC}$.

  Now consider modifying the choice of the premises of
  (\ref{strange-quasi-identity}) from our earlier choice,
  but keeping $u(x_1,x_2)=x_1$ and $v(x_1,x_2)=x_2$.  In
  this case, (\ref{strange-quasi-identity}) still asserts
  that the bad intervals are trivial, but the bad intervals
  need not be the $\sdm$-failures.  Instead, the bad
  intervals are those of the form $I[d,e]$ where for some
  $a, b, c\in L$ we have $w_i(a,b,c)=d$ for all $i$ and
  $e=a\wedge (b\vee c)$. A more direct way of asserting that
  $I[d,e]$ is trivial is to assert that $e=d$ (where
  $e=a\wedge(b\vee c)$).  This is achieved by rewriting
  quasi-identity (\ref{strange-quasi-identity}) in the form
  \begin{equation}\label{strange-quasi-identity2}
    \bigwedge_{i=1}^m \big(w_i(p,q,r)\approx s\big)
    \to \big((p\wedge(q\vee r))=s\big)\,.
  \end{equation}
  Quasi-identity (\ref{strange-quasi-identity2}) is
  equivalent modulo the identities defining lattice theory
  to quasi-identity (\ref{strange-quasi-identity}) when
  $u(x_1,x_2)=x_1$ and $v(x_1,x_2)=x_2$.  In particular,
  Theorem~\ref{strange-thm} proves that a quasi-identity
  obtained from the meet semidistributive law,
  \[
  \big((p\wedge q) \approx s\big) \wrel{\&} \big((p\wedge r)
  \approx s\big) \to \big((p\wedge(q\vee r))=s\big)\,,
  \]
  by adding additional premises $w_i(p,q,r)\approx s$ (where
  $w_i(p,q,r)\leq t$) is equivalent to a meet continuous
  identity.

  By examining the proof of Theorem~\ref{strange-thm} one
  finds that the bad intervals in $\m l$ associated to a
  quasi-identity of type (\ref{strange-quasi-identity}) or
  (\ref{strange-quasi-identity2}) are those of the form
  $I[d,e]$ where for some $a, b, c\in L$ it is the case that
  $d = \delta^{\wec{w}}(a,b,c)$ and $e=a\wedge (b\vee c)$.
  Since quasi-identity (\ref{strange-quasi-identity2})
  asserts only that bad intervals are trivial, a simpler
  meet continuous identity equivalent to
  (\ref{strange-quasi-identity2}) is
  \begin{equation}\label{strange-identity2}
    \delta^{\wec{w}}(p,q,r)\approx
    \big (p\wedge(q\vee r)\big)\,.
  \end{equation}
\end{rem}

It is not hard to describe explicitly the meet continuous
identity that is equivalent to the meet semidistributive
law.  Define lattice words in the variables $p$, $q$, $r$ by
$q_0 = q$, $r_0 = r$, $q_{n+1} = q\vee(p\wedge r_n)$, and
$r_{n+1} = r\vee(p\wedge q_n)$.  Let $q_{\omega} =
\bigvee_{n<\omega} q_n$ and let $r_{\omega} =
\bigvee_{n<\omega} r_n$.

\begin{cor}\label{strange-cor}
  A meet continuous lattice is meet semidistributive if and
  only if it satisfies the identity
  \begin{equation}\label{SD-meet-identity}
    p\wedge(q\vee r) \approx p\wedge q_{\omega}\,.
  \end{equation}
\end{cor}

\begin{proof}
  This is an instance of the comments in the final paragraph
  of Remark~\ref{sdm_rem} above.  For the choices
  $w_1(p,q,r)=p\wedge q$ and $w_2(p,q,r)=p\wedge r$ the
  polynomials $f_i$ that occur in the proof of
  Theorem~\ref{strange-thm} are $f_1(x)=p\wedge (q\vee
  x)=:g(x)$ and $f_2(x)=p\wedge (r\vee x)=:h(x)$.  Since
  these are idempotent polynomials of $\m f_{{\vr
      L}_{MC}}(p,q,r)$, the words of the form $f_{i_1}\circ
  \cdots \circ f_{i_{\ell}}(0)$ that occur in the preceding
  proof reduce to alternating compositions $(gh)^k(0)$,
  $(hg)^k(0)$, $g(hg)^k(0)$, or $h(gh)^k(0)$.  The join of
  these words is $\delta^{\wec{w}}(p,q,r) =
  \bigvee_{k<\omega} (gh)^k(0) = \bigvee_{k<\omega}
  (hg)^k(0)$. But $g(0)=p\wedge q_0$, $h(0) = p\wedge r_0$,
  $g(p\wedge r_n) = p\wedge q_{n+1}$ and $h(p\wedge q_n) =
  p\wedge r_{n+1}$, so $\delta^{\wec{w}}(p,q,r) =
  \bigvee_{\mbox{\rm\tiny $k$ odd}} p\wedge q_k = p\wedge
  q_{\omega}$ by meet continuity, and also
  $\delta^{\wec{w}}(p,q,r) = \bigvee_{\mbox{\rm\tiny $k$
      even}} p\wedge r_k = p\wedge r_{\omega}$.  Hence the
  meet semidistributive law is equivalent to $p\wedge
  q_{\omega}\approx p\wedge (q\vee r)$ or to $p\wedge
  r_{\omega}\approx p\wedge (q\vee r)$ for meet continuous
  lattices.
\end{proof}

This proof shows that $p\wedge q_{\omega}\approx p\wedge
r_{\omega}$, so identity (\ref{SD-meet-identity}) can be
written as a weakened distributive law:
\[
p\wedge(q\vee r)\approx (p\wedge q_{\omega}) \vee 
(p\wedge r_{\omega})\,.
\]
