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.

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