A variety V of finite (ordered) semigroups is a class of finite (ordered) semigroups that: 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.
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.
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
More examples are given in the article Special classes of semigroups.
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.
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
It is a subsemigroup of the monoid S1 formed by adjoining an identity element.
Reiterman's theorem states that the two definitions above are equivalent.
Given a variety V of semigroups as in the algebraic definition, one can choose the set P of profinite identities to be the set of profinite identities satisfied by every semigroup of V. Reciprocally, given a profinite identity u = v, one can remark that the class of semigroups satisfying this profinite identity is closed under subsemigroups, quotients, and finite products.
Furthermore, varieties are closed under arbitrary intersection, thus, given an arbitrary set P of profinite identities ui = vi, the class of semigroups satisfying P is the intersection of the class of semigroups satisfying all of those profinite identities.
We recall the definition of a variety in universal algebra.
In this section "variety of (arbitrary) semigroups" means "the class of semigroups as a variety of universal algebra over the vocabulary of one binary operator".
We first give an example of a variety of finite semigroups that is not similar to any subvariety of the variety of (arbitrary) semigroups.
We then give the difference between the two definition using identities.
Finally, we give the difference between the algebraic definitions.
However, the class of groups is not a subvariety of the variety of (arbitrary) semigroups.
Since the class of (arbitrary) groups contains a semigroup and does not contain one of its subsemigroups, it is not a variety.
While infinite groups are not closed under taking submonoids.
The class of finite groups is a variety of finite semigroups, while it is not a subvariety of the variety of (arbitrary) semigroups.
Thus, Reiterman's theorem shows that this class can be defined using profinite identities.
And Birkhoff's HSP theorem shows that this class can not be defined using identities (of finite words).
This illustrates why the definition of a variety of finite semigroups uses the notion of profinite words and not the notion of identities.
Requiring that varieties are closed under arbitrary direct products implies that a variety is either trivial or contains infinite structures.
In order to restrict varieties to contain only finite structures, the definition of variety of finite semigroups uses the notion of finite product instead of notion of arbitrary direct product.