MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $aNbN=abNN=abN.$

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.