In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups satisfying specific algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notions. Varieties of finite monoids, varieties of finite ordered semigroups and varieties of finite ordered monoids are defined similarly.

This notion is very similar to the general notion of varieties and pseudovarieties in universal algebra.

There are two standard equivalent definitions for a variety of finite semigroups.

A variety ${\displaystyle V}$ of finite (ordered) semigroups is a class of finite (ordered) semigroups that:

The first condition is equivalent to stating that ${\displaystyle V}$ is closed under taking subsemigroups and under taking quotients. The second property implies that the empty product—that is, the trivial semigroup of one element—belongs to each variety. Hence a variety is necessarily non-empty.

A variety of finite (ordered) monoids is a variety of finite (ordered) semigroups whose elements are monoids. That is, it is a class of (ordered) monoids satisfying the two conditions stated above.

In order to give the topological definition of a variety of finite semigroups, some other definitions related to profinite words are needed.

Let ${\displaystyle A}$ be an arbitrary finite alphabet. Let ${\displaystyle A^{+}}$ be its free semigroup. Then let ${\displaystyle {\hat {A}}}$ be the set of profinite words over ${\displaystyle A}$. Given a semigroup morphism ${\displaystyle \phi :A^{+}\to S}$, let ${\displaystyle {\hat {\phi }}:{\hat {A}}\to S}$ be the unique continuous extension of ${\displaystyle \phi }$ to ${\displaystyle {\hat {A}}}$.