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

Here we fix notation and introduce 
%elementary 
definitions and results
with the purpose of providing a bridge from
%the 
standard material 
%found in 
%\cite{birkhoff, freese-mckenzie, garcia-taylor, hobby-mckenzie, 
%maclane, mmt, taylor1}
to the material in this monograph.
%More detailed information may be found in the sources cited
%in the section titles.

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}
%\cite{hodges,mmt}
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 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 $\m a$.
%A {\bf partial algebra}\index{partial algebra} 
%of signature $\sigma$ is defined the same way,
%except that each $f^{\m a}$ may be a partial operation,
%which means that the domain of $f^{\m a}$ 
%need only be a subset $D\subseteq A^k$.

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 
$fg_1\cdots g_k\in T(Y)$.
\end{enumerate}
When $Y = X := \{x_i\;|\;1\leq i<\omega\}$, 
then $T(X)$ is the set
of {\bf terms of signature $\sigma$},
and its members are called {\bf terms}.\index{terms}
When $Y = X_n := \{x_1,\ldots,x_n\}$,
the members of $T(X_n)$ are {\bf $n$-ary terms}.

$T(Y)$ has a canonical structure
of an algebra of signature $\sigma$. 
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 
$f^{\m t}(g_1,\ldots,g_k) = fg_1\cdots g_k\in T$.
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 
$\widehat{v}\colon \m t\to \m a$; i.e., $\m t$ 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 $A$,
$t^{\m a}\colon A^n\to A$, defined by 
$(a_1,\ldots,a_n)\mapsto t(v)$ where
$v\colon X_n\to A$ is the assignment 
$x_i\mapsto a_i$. 
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 $n$.
$\m a$ {\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 $\m a$ 
%or class $\vr K$ of algebras 
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 {\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)] 
$F = \{\pi_i^n\wrel{|} n\in {\omega}, 1\leq i\leq n\}
\cup \{{\rm comp}_n^m\wrel{|} m, n\in {\omega}\}\rb$ 
is a set of operations between the $C_i$.
%(called {\bf projections} and {\bf compositions} respectively).
\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)] 
${\rm comp}_{m}^p(f,{\rm comp}_{m}^n(g_1, h_1,\ldots, h_n),
\ldots, {\rm comp}_{m}^n(g_p, h_1,\ldots, h_n)) = \\
{\rm comp}_{m}^n({\rm comp}_n^p(f,g_1, \ldots, g_p), h_1,\ldots, h_n)$.
\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 $x_i$,
%the $n$-ary projection onto the $i$-th variable,
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. 
%This puts
%us in position to define the clone of a variety.

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
%expressing that the constant $\pi_i^n\in C_n$ is the
%operation of $n$-ary projection onto the $i$-th variable, and
\item[(ii)] composition identities: 
\[
f(x_1,\ldots,x_n)\approx 
g(h_1(x_1,\ldots,x_n),\ldots,h_m(x_1,\ldots,x_n))
\]
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 
$\vr C = \Clo({\rm Var}({\vr C}))$ holds for any clone $\vr C$,
while $\vr V$ is definitionally equivalent to
the variety ${\rm Var}({\Clo(\vr V)})$ 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.
%For example, our statement 
%%the fact mentioned 
%in the Introduction that
%every Boolean algebra has an underlying group structure
%%means precisely 
%was intended to mean that $\vr G\leq \vr B$ where
%$\vr G$ represents the variety of groups
%and $\vr B$ represents the variety of Boolean algebras.


\section{Lattice Theory}
%\cite{birkhoff, freese-jezek-nation}}
\label{lattice_theory}
%(Citation: \cite{birkhoff})
%The primary language used in this monograph to 
%describe the shapes of congruence lattices
%is the language of 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 {\bf lattice}\index{lattice} 
is a total algebra $\m l=\lb L; \vee, \wedge\rb$ 
on which there is an order making 
$\lb L; \vee, \wedge, \leq\rb$ a partial 
lattice. 
%These definitions are consistent
%with our earlier definitions of ``algebra''
%and ``partial algebra''.

%The lattice operations are 
%binary join and meet,
%denoted $\vee$ and $\wedge$ respectively. 
If $x \leq y$ in some lattice $\m l$, 
then the {\bf interval}\index{interval} 
between them
is $I[x,y] := \{ z\wrel{|} x\leq z \leq y\}$.
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
close 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 $(x]:= \{z\wrel{|} z\leq x\}$.
The collection $\vr I(\m l)$ 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 
\[
I\vee J = \{z\wrel{|} \exists x\in I, y\in J (z\leq x\vee y)\}.
\]

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 $(\m p(Q),p\approx q)$
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}
((p\wedge q) \approx s)\wrel{\&} ((p\wedge r) \approx s)\to
((p\wedge (q \vee r)) \approx s),
\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 
%The meet semidistributive law may be written in the 
%following
%equivalent form:
\begin{equation}\label{sdm_equiv_version}
((p\wedge q)\approx s)\wrel{\&}
((p\wedge r)\approx s)\wrel{\&}
((q\vee r)\geq p)\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 
$(\m p(\sdm),p\approx m)$.
%\begin{df}
%\label{sdm_failure}\label{sdj_failure}
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 
%$I$ 
of 
%more than one element
%that has 
the form 
$I=I[\varphi(m),\varphi(p)]$ where
$\varphi\colon \m p(\sdm)\to \m l$ is
a homomorphism of partial lattices.
%$a, b, c\in L$ and $a\wedge b = a\wedge c$.
This $\sdm$-failure is {\bf trivial} if $I$
has one element and is {\bf nontrivial} otherwise.
{\bf The $\sdj$-configuration} and 
{\bf $\sdj$-failures}
%\index{SDJ@$\sdj$-failure}
%\index{failure!join semidistributive}
are defined dually.
%\end{df}
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)] (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.)
%\medskip

\begin{figure}[ht]
\begin{center}
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\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}}

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\put(90,25){\circle*{1.2}}
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\put(105,25){\circle*{1.2}}
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\put(120,25){\circle*{1.2}}

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\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}}

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\put(45,15){\line(0,1){10}}
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\put(105,25){\line(3,-2){15}}
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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}
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\put(5,20){\circle*{1.2}}
\put(15,5){\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}}
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\put(70,15){\circle*{1.2}}
\put(70,25){\circle*{1.2}}

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\put(15,5){\line(-2,3){10}}
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\put(60,5){\line(-2,3){10}}
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\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}
%\bigskip\bigskip

\item[(2)] 
(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
$$
(x\wedge y\leq u\vee v)\to
((x\leq u\vee v) \;{\rm or}\;
(y\leq u\vee v) \;{\rm or}\;
(x\wedge y\leq u) \;{\rm or}\;
(x\wedge y\leq v)).
$$
\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 
$\lb G \wrel{|} R\rb$ 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 
%whether or not 
a quasi-identity $Q$
satisfies $\W$ iff 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 $\W$.
%can be determined by the following algorithm:
%Let $\m p$ be the natural 
%partial lattice of subterms of 
%premises of $Q$. Then
%$Q$ satisfies $\W$ if
%and only if $\m p$ satisfies $\W$.
%The latter means that if $a, b, c, d \in P$,
%$a\wedge b$ and $c\vee d$
%are both defined, and 
%$a\wedge b\leq c\vee d$, then 
%$\{a, b, c, d\}\cap I[a\wedge b, c\vee d] \neq \emptyset$.
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 $\W$.
%Since $\W$ is self-dual, 
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
has a largest element, 
is {\bf lower bounded}\index{bounded!lower} if each 
kernel class $h^{-1}(a)$, $a\in L$, has a least element,
and is {\bf bounded}\index{bounded}
if it is both lower and upper bounded.
A 
%finitely generated 
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
$\sigma\circ \overline{\varphi} = \varphi$.

\begin{center}
\begin{figure}
$$
{\xymatrix{
& {\m p}\ar@{-->}[ld]_*+{\overline{\varphi}}\ar[d]^*+{\varphi}\\
  {\m A} \ar@{->>}[r]_*+{\sigma}   & {\m B}
}}
$$
\caption{ \ }\label{fig2}
\end{figure}
\end{center}
\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 a
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}

\begin{proof}
Let ${\vr U}$ consist of all 
members of $\vr V$
that have no subalgebra isomorphic to $\m p$.
$\vr U$ 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 call the identity $\varepsilon$ of
Theorem~\ref{proj+si_implies_splitting}
the {\bf conjugate identity}\index{conjugate identity}
for $\m p$.

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 $Q$. 
$\vr U$ 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}

%There is an extension of the language of lattice theory
%that will play a role in this monograph.
%, it
%is the language of meet continuous lattices.
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}
%In this monograph, meet continuity
%will always include the assumption of completeness.
Every algebraic lattice
%, in particular
%$\Conb(\m a)$ for any algebra $\m a$,
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.
%\footnote{Some
%new identities are needed in the infinitary case,
%e.g. $x\vee \bigvee (x\wedge y_i) \approx x$.}
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.
%rather than up-directed joins by replacing
%the join with the up-directed join
%of its finite sub-joins. 
This is done by by replacing
an arbitrary join with the up-directed supremum
of its finite sub-joins:
\[
x\wedge \bigvee_{y\in {\vr F}} y 
\wrel{\approx}
\bigvee_{
\begin{array}{c}
\mbox{\scriptsize $\vr F_0\subseteq \vr F$}\\
\mbox{\scriptsize $\vr F_0 \;\mbox{\rm finite}$}
\end{array}} 
\left(x\wedge \bigvee_{y\in {\vr F}_0} y\right).
\]

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 
$\vr I(h)\colon \vr I(K)\to \vr I(L)$
is defined by 
\[
\vr I(h)(I) = \{z\in L\wrel{|} \exists x\in I (z\leq h(x))\}.
\]

\begin{thm}\label{adjoint}
Restriction to $\m l$ is a natural bijection
\[
%\varphi\colon
\mid\colon
\vr L_{MC}(\vr I(\m l), \m k)\to \vr L(\m l, F(\m k)).\]
\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
\[
\widehat{g}\colon \vr I(\m l)\to\m k\colon
I\mapsto \bigvee \{ g(x)\wrel{|} x\in I\}
\]
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 
$\vr S\vr E\vr T\stackrel{free}{\longrightarrow} 
\vr L\stackrel{\vr I}{\longrightarrow} \vr L_{MC}$
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 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 $\{(x]\wrel{|} x\in X\}$.
\end{cor}


Since ${\vr L}_{MC}$ is definable by identities
%(in an infinitary language) 
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}%
%, \cite{garcia-taylor}}
\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 
%$\sigma = 
``$\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 
$\lb F\wrel{|}\Sigma\rb$ 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
\;\wrel{\&}\;
{\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 $\vr V$.
%, and we call $f$ an {\bf idempotent term of $\vr V$}.
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)] $k(x_{11}, \ldots, x_{n n}) \approx
h(h(x_{11}, \ldots, x_{1 n}), \ldots, h(x_{n1}, \ldots, x_{n n}))$, 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
$f(\mbox{\rm variables})\approx f(\mbox{\rm variables})$.
\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 $j\not \in U$.
Observe that any identity of the form
\[
f(\mbox{\rm variables})\approx f(\mbox{\rm variables})
\]
that uses only the variable $x$ 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 $U\equiv V$. 
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 $\vr F$.

%When we want to express 
%a 2-variable Maltsev condition of the type
%described in Lemma~\ref{id_linear}, then
%we will use $\vr B(f)$ 
%to encode the identities of type~\ref{id_linear}~(ii).

\begin{exmp} \label{maltsev_term}
The statement that $p$ is a Maltsev term for $\vr V$
%(cf.\ (\ref{cp_cond}))
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 
$p(y,y,x)\approx p(x,x,x)$.
In this case, 
$N = \{1, 2, 3\}$ 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}
equipped with an equivalence relation $E$
where 
$N\equiv_E \{1\}\equiv_E \{3\}$, since
$p(x,x,x)\approx p(x,y,y)\approx p(y,y,x)$
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{taylor_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, i.e., 
there is some $U\in \vr U$ and some 
$V\not\in \vr U$ such that $\vr V$
satisfies $f_U(x,y)\approx f_V(x,y)$.
Another way to express Condition~(3) is that,
for each $1\leq i\leq n$, $\vr V$
satisfies an identity of the type 
$f(\mbox{\rm variables})\approx f(\mbox{\rm variables})$
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 $f$.
%Explicitly, an $i$-th Taylor identity for $f$
%is an identity of the form 
%$f(x_{1},\ldots,x_{n})\approx 
%f(y_{1},\ldots,y_{n})$ where $\{x_i,y_i\}=\{x,y\}$ and
%$x_{j}, y_{j}\in \{x,y\}$ for all $j$.

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|(}
%\section{Congruences and the Term Condition}
%\section{Congruences, $\Conb(\m a)$ and the Term Condition}
%(Citation: \cite{hobby-mckenzie})

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 $\m a^n$.
%An operation $f\colon A^n\rightarrow A$
%is {\bf compatible} if its graph
%$
%\{(a_1,\ldots,a_n,f(a_1,\ldots,a_n))\wrel{|}
%(a_1,\ldots,a_n)\in A^n\}
%$
%is a compatible relation. This condition
%holds exactly when $f\colon \m a^n\to \m a$
%is a homomorphism. 
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) \wrel{|}(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 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 $\Cg a((a,b), (c,d))$ 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}
%\index{trivial!tolerance}
 if it is the equality relation
and {\bf nontrivial}
%\index{nontrivial!congruence}
%\index{nontrivial!tolerance}
 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 
$[S, T]\leq \delta$ 
%if and only if
%$S$ centralizes $T$ modulo $\delta$
(see Chapter~1 of \cite{freese-mckenzie}).
%This fact may make some of the following basic properties of 
%%$\C C({\hphantom{S}},{\hphantom{T}};{\hphantom{T}})$ 
%the centralizer relation plausible.

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)] 
%If $\alpha = \Cg a(S)$, then 
$\C C(S,T;\delta)$ 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
$\C C(S,T;\delta)$ holds if and only if 
$\C C(S,T;\delta')$ holds.
\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 $T\cap(S\circ (T\cap\delta)\circ S)
\subseteq\delta$, then $\C C(S,T;\delta)$ holds.
\item [(8)] 
If $\beta\wedge(\alpha\vee(\beta\wedge\delta))\leq\delta$,
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 $\m a$, 
%and $\m b$ is a subalgebra of $\m a$, 
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), 
$\C C(\Cg a(S),T;\delta)\mimplies \C C(S,T;\delta)$ 
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
${\rm tr.cl.}\left(\bigcup_{i\in I} S_i\right)$,
so 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 $M(S,T)$ are $T$-related. Therefore, 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 
$p\equiv_{\delta'} q \mimplies r\equiv_{\delta'} s$.

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 
$r\wrel{T\cap (S\circ (T\cap\delta)\circ S)} s$. 
By the assumption in (7), this implies that 
$r\equiv_{\delta} s$.  This proves (7).

For item (8), if
$\beta\wedge(\alpha\vee(\beta\wedge\delta))\leq\delta$,
then 
$\beta\cap(\alpha\circ(\beta\cap\delta)\circ \alpha)\leq\delta$,
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 $\{\Conb(\m a)\wrel{|}\m a\in \vr V\}$
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 $\{\Conb(\m a)\wrel{|}\m a\in \vr V\}$,
or alternatively is the variety 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}\label{czedli-day-W}
(Cf.\ \cite{czedli-day})
Let $Q$ be a quasi-identity satisfying $\W$.
The class of varieties satisfying $Q$
as a congruence quasi-identity 
%whose congruence lattices satisfy $Q$ 
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
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 (w_i(p,q,r)\approx s)\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}
For each $i$ let $f_i(x)$
be the polynomial $w_i(p,q\vee x,r\vee x)$
of the free meet continuous 
lattice $\m f_0=\m f_{{\vr L}_{MC}}(\{p,q,r\})$.
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}{rl}
\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((q\vee r)\vee (q\vee r)) = 
\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 
$\varphi(\delta^{\wec{w}}(p,q,r)) = 
\delta^{\wec{w}}(a,b,c)$ 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 used 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 
%This implies that $\delta^{\wec{w}}(a,b,c)$ is 
the least
common fixed point of the polynomials $f_i^{\m l}$
in $\m l$.
%since updirected suprema are
%preserved by meet continuous homomorphisms.

For each variable $x_i$, set $x_i^{+}:=
(x_i\vee \delta^{\wec{w}}(p,q,r))\wedge t$,
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}
%\delta^{\wec{t}}(p,q,r)\approx p\wedge (q\vee r)
%\end{equation}
\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 
$$
(\psi(p),\psi(q),\psi(r),\psi(s),\psi(\wec{x}))=
(a,b,c,d,\wec{g}),
$$
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 $w_k(p,q,r)\leq (p\wedge(q\vee r))\leq p$ for all $k$
in $\m f_0$,
it follows that 
$$d=w_i(a,b,c)=w_j(a,b,c)=w_k(a,b,c)\leq 
%b\wedge c\wedge (a\wedge(b\vee c))= 
a\wedge b\wedge c
$$
in $\m l$.
Since $d=w_1(a,b,c)\geq a\wedge b\wedge c$ we even
have $d = a\wedge b\wedge c$.
%$,
%d=w_j(a,b,c)\leq c$, and 
%$d=w_k(a,b,c)\leq a$, so 
%$d\leq a\wedge b\wedge c$.
Therefore 
$f_i^{\m l}(d)=t^{\m l}_i(a,b\vee d,c\vee d) = 
t^{\m l}_i(a,b,c) = d$ for all $i$,
yielding that $d$ is common fixed point 
the $f_i$ in $\m l$. 
%By the conclusion
%of the previous paragraph, 
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
$$
u^{\m l}(\wec{g}^*)
=
\psi(u(\wec{x}^+))=
\psi(v(\wec{x}^+))=
v^{\m l}(\wec{g}^*), 
$$
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 $\psi(u(\wec{x}^+))=
\psi(v(\wec{x}^+))$.
Since the variable $s$ does not appear
(\ref{strange-identity}) we need only
consider homomorphisms $\psi$
for which $\psi(s)=\psi(\delta^{\wec{w}}(p,q,r))$,
i.e., those of the form
$$
(\psi(p),\psi(q),\psi(r),\psi(s),\psi(\wec{x}))=
(a,b,c,\delta^{\wec{w}}(a,b,c),\wec{g}).
$$
Under such homomorphisms 
$$\psi(x_i^+)=\psi(x_i^*)
=(g_i\vee \delta^{\wec{w}}(a,b,c))\wedge (a\wedge(b\vee c)).$$

%For any three elements $a, b, c\in L$,
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}(\psi^{\delta}(\wec{x}^*)) = 
v^{\m l}(\psi^{\delta}(\wec{x}^*))
\end{equation}
holds in $\m l$, where 
%$\overline{\psi}(x_i^*) = 
%\overline{\psi}((x_i\vee s)\wedge t) = 
%(g_i\vee \delta)\wedge \overline{\psi}(t)$.
%Recall that, in the definition of $x_i^*$,
%$t = p\wedge (q\vee r)$; therefore 
$$\psi^{\delta}(x_i^*) = 
(g_i\vee \delta)\wedge (a\wedge(b^{\delta}\vee c^{\delta})).$$
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 
$$
a\wedge(b\vee c)\leq 
a\wedge(b^{\delta}\vee c^{\delta}) = 
a\wedge(b\vee c \vee {\delta}) 
=
a\wedge(b\vee c).
$$
%The first inequality follows from the
%monotonicity
%of the lattice operations.
This shows that $a\wedge(b^{\delta}\vee c^{\delta}) 
= a\wedge(b\vee c)$, so 
$\psi^{\delta}(x_i^*) = 
(g_i\vee \delta)\wedge (a\wedge(b\vee c)) = \psi(x_i^+).$
From this and (\ref{uv_eqn}) we derive that
$$
\psi(u(\wec{x}^+)) = 
%u(\psi(\wec{x}^+)) = 
u^{\m l}(\psi^{\delta}(\wec{x}^*)) = 
v^{\m l}(\psi^{\delta}(\wec{x}^*)) =
%v(\psi(\wec{x}^+)) =
\psi(v(\wec{x}^+)),
$$
as desired.
This proves that (\ref{strange-identity}) holds in $\m l$.
\end{proof}

\begin{rem}\label{sdm_rem}
If $\m l$ is meet continuous and
satisfies quasi-identity 
(\ref{strange-quasi-identity}),
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 $a\wedge (b\vee c)=:e$.
Call $I:=I[d,e]$ a ``bad interval'' in $\m l$
if it arises in this way from elements
satisfying the premises of (\ref{strange-quasi-identity}). 
For each $i$, 
$\psi(x_i^*) = (g_i\vee d)\wedge e\in I$, 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
%that there is a meet continous identity 
%identity defining
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 (w_i(p,q,r)\approx s)\to ((p\wedge(q\vee r))=s)\,.
\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,
$$
((p\wedge q) \approx s) \&
((p\wedge r) \approx s) \to 
((p\wedge(q\vee r))=s),
$$
by adding additional premises of the form $w_i(p,q,r)\approx s$
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 (p\wedge(q\vee r))\,.
\end{equation}
\end{rem}

%The specific choices of $w_i$ in Remark~\ref{sdm_rem}
%allow us to 
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}.
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:
\[
p\wedge(q\vee r)\approx (p\wedge q_{\omega}) \vee 
(p\wedge r_{\omega})\,.
\]

%Theorem~\ref{strange-thm} will be applied
%to a different quasi-identity 
%in Theorem~\ref{typ1condcor}.

%$\sdm/\Mod$ factorization ***define somewhere*** 


