\chapter{Varieties With Special Terms}\label{appendix}

Throughout the text there are theorems
characterizing the two classes of varieties 
(i) possessing a
Taylor term or (ii) possessing a Hobby-McKenzie term.
They are collected here.

\bigskip
\bigskip

\section{Varieties With a Taylor Term}
%\centerline{\bf Varieties With a Taylor Term}

\bigskip

\begin{thm}\label{appendix_taylor}
A variety $\vr V$ has a Taylor term\footnote{Definition~\ref{taylor_term}.}
if and only if
one of the following conditions is satisfied.
\begin{enumerate}
\item[(1)]
$\vr V$ satisfies an idempotent Maltsev condition that fails
in some variety. (Thm.~\ref{wtaylor_term})
\medskip

\item[(2)]
$\vr V$ satisfies an idempotent Maltsev condition that fails
in the variety of sets. (Thm.~\ref{wtaylor_term})
\medskip

\item[(3)]
$\vr V$ has no member with a nonzero strongly abelian congruence.
(Thm.~\ref{str_ab_thm1})
\medskip

\item[(4)]
$\vr V$ has no member with a nonzero strongly abelian tolerance.
(Thm.~\ref{str_ab_thm1})
\medskip

\item[(5)]
There exist $m\ge 1$, $n\ge 1$, 
idempotent sixary terms $f_1,\ldots,f_m$,
$g_1,\ldots,g_n$
and a
binary term $s$ such that the following identities hold in $\vr V$:
\begin{enumerate}
\item[(i)] $s(x, y) \approx f_1(y, y, x, x, x, y)$,
\item[(ii)] $f_m(y, x, x, x, y, y) \approx x$,
\item[(iii)] $f_i(x, x, x, y, y, y) \approx f_i(y, y, x, x, x, y)$, 
$1 \le i \le m$,
\item[(iv)] 
$f_i(y, x, x, x, y, y) \approx f_{i+1}(y, y, x, x, x, y)$,
$1 \le i \le m-1$,
\item[(v)] 
$s(x, y) \approx g_1(y, y, x, x, x, y)$,
\item[(vi)] 
$g_n(y, x, x, x, y, y) \approx y$,
\item[(vii)] 
$g_i(x, x, x, y, y, y) \approx g_i(y, y, x, x, x, y)$, 
$1 \le i \le n$,
\item[(viii)] 
$g_i(y, x, x, x, y, y) \approx g_{i+1}(y, y, x, x, x, y)$, 
$1 \le i \le n-1$. (Thm.~\ref{omit_1})
\end{enumerate}
\medskip

\item[(6)]
$\vr V$ has a join term. (Thm.~\ref{omit_1})
\medskip

\item[(7)]
$\vr V$ has no member with a nonzero strongly rectangular congruence.
(Thm.~\ref{str_ab_thm2})
\medskip

\item[(8)]
$\vr V$ has no member with a nonzero strongly rectangular tolerance.
(Thm.~\ref{str_ab_thm2})
\medskip

\item[(9)]
The quasi-identity
\[
      \big((p\wedge q)\approx s\big)
      \wrel{\&} \big((p\wedge r)\approx s\big)
      \wrel{\&} (p_{[2]}\approx s)
      \wrel{\to} \big((p\wedge (q\vee r))\approx s\big)
\]
holds in the congruence lattices of algebras
    in $\vr v$.\footnote{The lattice word\/~$p_{[2]}$ is defined
    on page~\pageref{triangle-terms}.} (Thm.~\ref{typ1condcor})
\medskip

\item[(10)] 
The quasi-identity of Item (9)
holds in the meet
    continuous congruence variety of\/ $\vr V$. (Thm.~\ref{typ1condcor})
\medskip

\item[(11)] 
$\vr V$ satisfies a nontrivial meet continuous
    congruence identity. (Thm.~\ref{typ1condcor})
\medskip

\item[(12)] 
$\vr V$ satisfies a congruence inclusion of the
    form
    \[
    \alpha\cap(\beta\circ_{4}\gamma)\subseteq
    w(\alpha, \beta, \gamma)
    \]
    for some lattice word $w(p,q,r)$ such that
    $w(p,q,r)<p\wedge (q\vee r)$ in the free lattice
    $\m f_{\vr L}(\{p,q,r\})$. (Thm.~\ref{typ1condcor})
\medskip

\item[(13)] 
For some $k\geq 4$, $\vr V$ satisfies a
    congruence inclusion of the form
    \[
    \alpha\cap(\beta\circ_{k}\gamma)\subseteq w(\alpha,
    \beta, \gamma)\,,
    \]
    where $w(p,q,r) < p\wedge (q\vee r)$ in\/ $\m f_{\vr L}(\{p,q,r\})$.
    (Thm.~\ref{typ1condcor})
\medskip

\item[(14)]
$\vr V$ satisfies a congruence inclusion of the
    form
    \[
    \alpha\cap(\beta\circ_{3}\gamma)\subseteq w(\alpha,
    \beta, \gamma)\,,
    \]
    where $w(p,q,r)$ is a lattice word such that
    $w(p,q,r)\not\geq p^{n}$ for any $n$ in the free lattice
    $\m f_{\vr L}(\{p,q,r\})$.\footnote{The lattice word\/~$p^{n}$ is defined
    on page~\pageref{prep_4_7}.} (Proved in the remarks after 
    Thm.~\ref{typ1condcor})
\medskip

\item[(15)]
  For any finitely presented lattice\/ $\m l$ 
  satisfying $\W$ but failing
  quasi-identity (\ref{weak_W}), $\m l$ does not appear as a 
  sublattice of\/
  $\Conb(\m a)$ for any $\m a\in {\vr V}$.
  (For example, $\m l$~could
  be\/ $\m d_1$, $\m e_1$ or\/~$\m g$.\footnote{%
  These lattices are first defined in Theorem~\ref{not_jonsson_rival}.})  
   (Thm.~\ref{omit_d1_sub})
\medskip

\item[(16)]
  For any finitely presented lattice\/ $\m l$ 
  satisfying $\W$ but failing
  quasi-identity (\ref{weak_W}),   
  $\m l$ is not in the meet continuous congruence
  variety of $\vr V$.
  (Thm.~\ref{omit_d1_sub})
\medskip

\item[(17)]
  $\vr V$ has no member with a nonzero congruence that is 
  simultaneously rectangular and abelian.
  (Cor.~\ref{str_ab_thm3})
\medskip

\item[(18)]
  $\vr V$ has no member with a nonzero tolerance that is 
  simultaneously rectangular and abelian.
  (Cor.~\ref{str_ab_thm3})
\end{enumerate}
\end{thm}

\bigskip
\bigskip

\section{Varieties With a Hobby-McKenzie Term}
%\centerline{\bf Varieties With a Hobby-McKenzie Term}

\bigskip

\begin{thm}\label{appendix_HM}
A variety $\vr V$ has a Hobby-McKenzie 
term\footnote{Definition~\ref{hm_term}.} 
if and only if
one of the following conditions is satisfied.
\begin{enumerate}
\item[(1)]
    $\vr V$ satisfies an idempotent Maltsev
    condition that fails in the variety of semilattices.
    (Thm.~\ref{hobby-mckenzie_term})
\medskip

\item[(2)]
    No member of $\vr V$ has a nontrivial rectangular tolerance.\linebreak
    (Thm.~\ref{rect_Malt_cond2})
\medskip

\item[(3)]
    No member of $\vr V$ has a nontrivial rectangular congruence.\linebreak
    (Thm.~\ref{rect_cong})
\medskip

\item[(4)]
%\ref{rect_Malt_cond1}
    $x\sim_T y$ in\/ $\m f = \m f_{\vr V}(x,y)$
    for\/ $T = \Tg f (x,y)$.\footnote{The relations
    $\sim_T$ and $\ge_T$ are defined on page~\pageref{ge_defn}.} 
    (Thm.~\ref{rect_Malt_cond1})
\medskip

\item[(5)] There exists an $m\ge 1$, and sixary terms
    $f_1,\dots,f_m$ such that\/ $\vr V$ satisfies the
    following identities:
    \begin{enumerate}
    \item[(i)] $x\approx f_1(y,y,x,x,x,y)$,
    \item[(ii)] $f_m(y,x,x,x,y,y)\approx y$,
    \item[(iii)] $f_{i}(x,x,x,y,y,y) \approx
      f_{i}(y,y,x,x,x,y)$, $1\le i\le m$,
    \item[(iv)] $f_i(y,x,x,x,y,y)\approx
      f_{i+1}(y,y,x,x,x,y)$, $1\le i \le m-1$. (Thm.~\ref{rect_Malt_cond1})
    \end{enumerate}
\medskip

\item[(6)] For every algebra $\m a\in {\vr V}$, tolerance
    $S$ on $\m a$, and $(a,b)\in S$, $a\ge_S b$ holds.
    (Thm.~\ref{rect_Malt_cond1})
\medskip

\item[(7)] For every algebra $\m a\in {\vr V}$, tolerance
    $S$ on $\m a$, and $(a,b)\in S$, $a\sim_S b$ holds. 
    (Thm.~\ref{rect_Malt_cond1})
\medskip

\item[(8)]
    Whenever $\theta$, $\mu$, $\nu$ and $\delta$
    are congruences on some $\m a\in {\vr V}$, then
    $(\theta\circ\mu)\cap (\nu\circ\delta) \subseteq
    [(\theta\vee\delta)\wedge(\nu\vee\mu)] \vee
    \delta\vee\mu$. (Thm.~\ref{equiv})
\medskip

\item[(9)] $\vr V$ has a sequence of terms $f_i(x,y,u,v)$
    for $0\leq i\leq 2m+1$, such that
    \begin{enumerate}
    \item[(i)] ${\vr V}\models f_0(x,y,u,v)\approx x$;
    \item[(ii)] ${\vr V}\models f_i(x,y,y,y)\approx
      f_{i+1}(x,y,y,y)$ for even $i$;
    \item[(iii)] ${\vr V}\models f_i(x,x,y,y)\approx
      f_{i+1}(x,x,y,y)$ and
    \item[] ${\vr V}\models f_i(x,y,x,y)\approx
      f_{i+1}(x,y,x,y)$ for odd $i$;
    \item[(iv)] ${\vr V}\models f_{2m+1}(x,y,u,v)\approx v$.
    \end{enumerate}
     (Thm.~\ref{equiv})
\medskip

\item[(10)] 
    $\vr V$ omits special $\m d_2$'s. (Thm.~\ref{equiv})
\medskip

\item[(11)]
    ${\vr V}$ satisfies a nontrivial congruence
    identity. (Thm.~\ref{main1})
\medskip

\item[(12)] 
    There exists $\ell < \omega$ such that the
    lattice\/ $\N {\ell+5}$ cannot be embedded into\/
    $\Conb(\m a)$ for any $\m a\in {\vr V}$. (Thm.~\ref{main1})
\medskip

\item[(13)]
    There is a positive integer $N$ such that no
    algebra in~$\vr V$ has a herringbone of length greater
    than $N$ in its congruence lattice. (Thm.~\ref{main2})
\medskip

\item[(14)] 
    $\vr V$ satisfies the congruence identity
    \[
    z^M\approx z^{M+1}
    \]
    for some $M$.\footnote{The word $z^M$ is defined on 
    page~\pageref{main2}.} (Thm.~\ref{main2})
\medskip

\item[(15)]
    For every\/ $\m a\in {\vr V}$,\, $\Conb(\m a)$
    has an\/ $\sdj/\Mod$ factorization in the category of
    meet continuous lattices. (Thm.~\ref{SD-join/Mod})
\medskip

\item[(16)] 
    For every\/ $\m a\in {\vr V}$,\, $\Conb(\m a)$
    has an\/ $\sdj/\Mod$ factorization in the category of
    ordinary lattices. (Thm.~\ref{SD-join/Mod})
\medskip

\item[(17)] 
    For every\/ $\m a\in {\vr V}$, each\/
    $\sdj$-failure in\/~$\Conb(\m a)$ is modular. (Thm.~\ref{SD-join/Mod})
\medskip

\item[(18)] 
    If\/ $\m l$ belongs to\/ $\CON({\vr V})$, then
    each\/ $\sdj$-failure in\/~$\m l$ is modular. (Thm.~\ref{SD-join/Mod})
\medskip

\item[(19)] $\m d_2$ does not appear as a sublattice
    of\/~$\Conb(\m a)$ for any\/ $\m a\in {\vr V}$. (Thm.~\ref{omit_d2_sub})
\medskip

\item[(20)] $\m d_2\notin \CON(\vr V)$. (Thm.~\ref{omit_d2_sub})
\medskip

\item[(21)] The quasi-identity
\[
      \big((p\vee q)\approx s\big) 
      \wrel{\&}
      \big((p\vee r)\approx s\big) 
      \wrel{\&}
      \big(p^{[2]}\approx s\big) 
      \to
      \big((p\vee (q\wedge r))\approx s\big)
 \]
    holds in the congruence lattices of algebras in~$\vr v$.\footnote{%
    This quasi-identity is the dual of the one in 
    Theorem~\ref{appendix_taylor}~(9).}
    (Thm.~\ref{typ1condcor_dual})
\medskip

\item[(22)] 
    The quasi-identity of Item (20) holds
    in\/~$\CON({\vr V})$.
    (Thm.~\ref{typ1condcor_dual})
\medskip

\item[(23)] ${\vr V}\models_{con}
    \alpha\cap(\beta\circ\gamma) \subseteq
    \big(\gamma\vee(\alpha\wedge\beta)\big)\circ
    \big(\beta\vee(\alpha\wedge\gamma)\big)$.
    (Thm.~\ref{cong_id2})
\medskip

 \item[(24)] There is a positive integer $k$ and ternary
    terms $d_0,\ldots, d_{2k+1}$, $e_0,\ldots, e_{2k+1}$,
    $p$ such that $\mathcal V$ satisfies the following
    equations:
    \begin{enumerate}
    \item[(i)] $d_0(x,y,z)\approx p(x,y,z) \approx
      e_0(x,y,z)$;
    \item[(ii)] $d_{i}(x,y,y)\approx d_{i+1}(x,y,y)$ and
      $e_{i}(x,x,y)\approx e_{i+1}(x,x,y)$ for even $i$;
    \item[(iii)] $d_{i}(x,x,y)\approx d_{i+1}(x,x,y)$,
      $d_{i}(x,y,x)\approx
      d_{i+1}(x,y,x)$,
    \item[] $e_{i}(x,y,y)\approx e_{i+1}(x,y,y)$ and
      $e_{i}(x,y,x)\approx e_{i+1}(x,y,x)$ for odd~$i$;
    \item[(iv)] $d_{2k+1}(x,y,z)\approx x$ and
      $e_{2k+1}(x,y,z)\approx z$.
    \end{enumerate}
    (Thm.~\ref{cong_id2})
\end{enumerate}
\end{thm}


