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 $aNbN=abNN=abN.$



What are the semigroups in which congruence classes can be multplied like sets?
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.


