2. Some theorems showing Mal'tsev conditions

In addition to Mal'tsev's theorem, the following facts hold, among others. Some of them refer to a majority term, which means a ternary term $m(x,y,z)$ obeying the three laws

$m(x,x,y) = x$, $m(x,y,x) = x$, $m(y,x,x) = x$

in the variety in question.

2.1 Theorem (Pixley) For a variety $V$, the following are equivalent:

(a) $V$ is arithmetic;

(b) there are terms $p(x,y,z)$ and $m(x,y,z)$ such that in $V$, $p$ obeys Mal'tsev's laws of (b) in Theorem . and $m$ is a majority term;

(c) there is a term $q(x,y,z)$ such that in $V$,

$q(x,x,z) = z$ (minority of entries),

$q(x,z,z) = x$ (minority of entries),

$q(x,y,x) = x$ (majority of entries).

2.2 Theorem (Jónsson) For a variety $V$, the following are equivalent:

(a) $V$ is congruence-distributive;

(b) for some $n\geq 2$, there are terms $t_0,\dots, t_n$ in $x,y,z$ such that in $V$,

(i) $t_0 (x,y,z) = x$, $t_n (x,y,z) = z$;

(ii) $t_i (x,y,x) = x$, for all $i$;

(iii) $t_i (x,x,z) = t_{i+1} (x,x,z)$ for $i$ even, $t_i (x,z,z) = t_{i+1} (x,z,z)$ for $i$ odd.

(Notice that the case $n=2$ is equivalent to the existence of a majority term.)

2.3 Theorem (Day, Gumm) For a variety $V$, the following are equivalent:

(a) $V$ is congruence-modular;

(b) for some $n \geq 0$, there are terms $t_0,\dots, t_n$ and $p$ in $x,y,z$ such that in $V$,

(i) $t_0 (x,y,z) = x$

(ii) $t_i (x,y,x) = x$, for all $i$;

(iii) $t_i (x,z,z) = t_{i+1} (x,z,z)$ for $i$ even, $t_i (x,x,z) = t_{i+1} (x,x,z)$ for $i$ odd.

(iv) $t_n (x,z,z) = p(x,z,z)$,

(v) $p(x,x,z) = z$.