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For a semigroup $S$ and a congruence $\rho$ on $S$, let's say that $\rho$ is good when for all $a,b\in S$ we have that $[ab]=[a][b],$ where $[x]$ denotes the congruence class of $x$ modulo $\rho$ and the product on the right-hand side is the product of sets: $AB=\lbrace ab\,|\,a\in A,\ b\in B\rbrace$.

What are the semigroups whose all congruences are good?

Clearly, we always have $[a][b]\subseteq[ab]$ since if $a\rho x$ and $b\rho y$ then $(ab)\rho (xy)$ by the definition of a congruence. The other inclusion, however, doesn't always hold. Let $S=(\mathbb N,+)$ (with $0$) and let $\rho$ be the congruence modulo $2$. Then $[1+1]=2\mathbb N$ and $[1]+[1]=2\mathbb N\setminus \lbrace0\rbrace.$

It is true for groups though. For if $G$ is a group and $N$ a normal subgroup, then $aNbN=abNN=abN.$

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up vote 4 down vote accepted

These semigroups are called perfect. They were studied in:

Fortunatov, V.A. Perfect semigroups. (Russian) Izv. Vyssh. Uchebn. Zaved., Mat. 1972, No.3(118), 80-89.

The class of perfect semigroups is closed under homomorphisms and includes all completely (0)-simple semigroups.

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