## An equivalent characterization of semigroup congruences

Let $S$ be a semigroup and let $\sigma$ be an equivalence on $S$. Show that $\sigma$ is a congruence on $S$ if and only if for all $a,b,c,d \in S$, if $a \sigma b$ and $c \sigma d$, then $ac \sigma bd$.

Recall that an equivalence on $S$ is a congruence if for all $a,b,c \in S$, if $a \sigma b$ then $ac \sigma bc$ and $ca \sigma cb$.

Suppose $\sigma$ is a congruence, and suppose $a \sigma b$ and $c \sigma d$. Certainly $c \sigma c$ and $b \sigma b$, so we have $ac \sigma bc$ and $bc \sigma bd$. So $ac \sigma bd$.

Conversely, if $a \sigma b$, then we have $c \sigma c$, so that $ac \sigma bc$ and $ca \sigma cb$.