Algebraic definition

A variety V of finite (ordered) semigroups is a class of finite (ordered) semigroups that:

• is closed under division.
• is closed under taking finite Cartesian products.

The first condition is equivalent to stating that 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.

Topological definition

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

Let A be an arbitrary finite alphabet. Let A⁺ be its free semigroup. Then let ${\displaystyle {\hat {A}}}$ be the set of profinite words over 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 ${\displaystyle {\hat {A}}}$.

A profinite identity is a pair u and v of profinite words. A semigroup S is said to satisfy the profinite identity u = v if, for each semigroup morphism ${\displaystyle \phi :A^{+}\to S}$, the equality ${\displaystyle {\hat {\phi }}(u)={\hat {\phi }}(v)}$ holds.

A variety of finite semigroups is the class of finite semigroups satisfying a set of profinite identities P.

A variety of finite monoids is defined like a variety of finite semigroups, with the difference that one should consider monoid morphisms ${\displaystyle \phi :A^{*}\to M}$ instead of semigroup morphisms ${\displaystyle \phi :A^{+}\to M}$.

A variety of finite ordered semigroups/monoids is also given by a similar definition, with the difference that one should consider morphisms of ordered semigroups/monoids.

Using finite identities

• The most trivial example is the variety S of all finite semigroups. This variety is defined by the empty set of profinite equalities. It is trivial to see that this class of finite semigroups is closed under subsemigroups, finite products, and quotients.
• The second most trivial example is the variety 1 containing only the trivial semigroup. This variety is defined by the set of profinite equalities {x = y}. Intuitively, this equality states that all elements of the semigroup are equal. This class is trivially closed under subsemigroups, finite products, and quotients.
• The variety Com of commutative finite semigroups is defined by the profinite equality xy = yx. Intuitively, this equality states that each pair of elements of the semigroup commutes.
• The variety of idempotent finite semigroups is defined by the profinite equality xx = x.

More generally, given a profinite word u and a letter x, the profinite equality ux = xu states that the set of possible images of u contains only elements of the centralizer. Similarly, ux = x states that the set of possible images of u contains only left identities. Finally ux = u states that the set of possible images of u is composed of left zeros.

Using profinite identities

Examples using profinite words that are not finite are now given.

Given a profinite word, x, let ${\displaystyle x^{\omega }}$ denote ${\displaystyle \lim _{n\to \infty }x^{n!}}$. Hence, given a semigroup morphism ${\displaystyle \phi :A^{+}\to S}$, ${\displaystyle {\hat {\phi }}(x^{\omega })}$ is the only idempotent power of ${\displaystyle \phi (x)}$. Thus, in profinite equalities, ${\displaystyle x^{\omega }}$ represents an arbitrary idempotent.

The class G of finite groups is a variety of finite semigroups. Note that a finite group can be defined as a finite semigroup, with a unique idempotent, which in addition is a left and right identity. Once those two properties are translated in terms of profinite equality, one can see that the variety G is defined by the set of profinite equalities ${\displaystyle \{x^{\omega }=y^{\omega }{\text{ and }}x^{\omega }y=yx^{\omega }=y\}.}$

Classes that are not varieties

Note that the class of finite monoids is not a variety of finite semigroups. Indeed, this class is not closed under subsemigroups. To see this, take any finite semigroup S that is not a monoid. It is a subsemigroup of the monoid S¹ formed by adjoining an identity element.