Variety of finite semigroups

In mathematics, and more precisely in semigroup theory, a variety of finite semigroups is a set of semigroups having some nice algebraic properties. Those sets can be defined in two distinct way, using either algebraic notions or topological notions. Varieties of finite monoid, varieties of finite ordered semigroup and varieties of finite ordered monoid are defined similarly.

This notion is very similar to the general notion of variety in universal algebra.

Definition

Two equivalent definitions are now given.

Algebraic definition

A variety V of finite (ordered) semigroup is a set of finite (ordered) semigroups which:

is closed under division.
is closed under taking finite Cartesian product.

The first condition is equivalent to stating that V is closed under taking subsemigroup and under taking quotient The second property implie 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) monoid is a variety of finite (ordered) semigroup whose elements are monoid. That is, it is a set of (ordered) monoid satisfying the two conditions stated above.

Topological definition

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

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

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

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

A variety of finite monoid is defined as a variety of finite semigroup, with the exception that one should consider monoid morphism $\phi :A^{*}\to S$ instead of semigroup morphisms $\phi :A^{+}\to \phi$ .

A variety of finite ordered semigroups/monoids satisfies the same definition, with the exception that one should considered morphisms of ordered semigroup/monoid.

Examples

A few examples of set of semigroups are given. The first examples uses finite identities - that is, profinite identities whose two words are finite words. The next example use profinite identities. The last one is an example of set which is not a variety.

More examples are given in the article Special classes of semigroups.

Using finite identities

The most trivial examples is the variety S of every finite semigroups. This variety is defined by the empty set of profinite equalities. It is trivial to see that this set of finite semigroup is closed under finite product and quotient.
The second most trivial example is the variety 1 containing only the trivial semigroup. This variety is defined by the set of profinite equality {x = y}. Intuitively, this equally states that each elements of the semigroup are equals. This set is vacuously closed under finite product and quotient.
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 commute.
The variety of idempotent finite semigroup 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 images of u contains only elements of the centralizer. Similarly, ux = x states that the image of u contains left identities. Finally ux = u states that the image of u is composed of left zeros.

Using profinite identities

Examples using profinite words which are not finite is now given.

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

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

Sets which are not varieties

Note that the set of finite monoid is not a variety of finite semigroup. Indeed, this set is not closed under quotient. For example, takes the monoid of non-negative number with addition. The set of positive number with addition is a semigroup quotient of this monoid and it is not a monoid.

Reiterman's theorem

Reiterman's theorem states that the two definitions above are equivalent. A scheme of the proof is now given.

Given a variety V of semigroup 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, givens a profinite identity u = v, one can remark that the set of semigroups satisfying this profinite identity is closed under quotient and finite product. Thus this set is a variety of finite semigroups. Furthermore, varieties are closed under arbitrary intersection, thus, given an arbitrary set P of profinite identity uⁱ = vⁱ, the set of semigroup satisfying P is the intersection of the set of semigroups satisfying all of those profinite identities. That is, it is an intersection of varieties of finite semigroup, and this a variety of finite semigroup.

Comparison with the notion of variety of universal algebra

The definition of variety of finite semigroup is inspired by the notion of variety of universal algebra. We recall the definition of a variety in universal algebra. Such a variety is, equivalently:

a set of structure, closed under homomorphic images, subalgebras and (direct) products.
a set of structures satisfying a set of identities.

The main difference between the two notions of variety are now given. In this section "variety of (arbitrary) semigroup" means "the set of semigroup as a variety of universal algebra over the vocabulary of one binary operator". It follows from the definitions of those two kind of varieties that, for any variety V of (arbitrary) semigroups, the set of finite semigroups of V is a variety of finite semigroups.

We first give an examples of variety of finite semigroup which 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.

As shown above, the set of finite group is a variety of finite semigroup. However, the set of group is not a subvariety of the variety of (arbitrary) semigroups. Indeed, $\langle \mathbb {Z} ,+\rangle$ is a monoid which is an infinite group. However, its submonoid $\langle \mathbb {N} ,+\rangle$ is not a group. Since the set of (arbitrary) groups contains a semigroup and does not contain one of its subsemigroup, it is not a variety. The main difference between the finite case and the infinite case, when groups are considered, is that a submonoid of a finite group is a finite group. While infinite groups are not closed under taking submonoid.

The set of finite group is a variety of finite semigroups, while it is not a subvariety of the variety of (arbitrary) semigroups. Thus, Reiterman theorem show that this set can be defined using profinite identities. And Birkhoff's HSP theorem show that this set can not be defined using identities (of finite words). This illustrates why the definition of variety of finite semigroups use the notion of profinite words and not the notion of identities.

We now consider the algebraic definitions of varieties. Requiring that varieties are closed under arbitrary direct product implies that a variety is either trivial or contains infinite structures. In order to restrict varieties to contains only finite structures, the definition of variety of finite semigroups uses the notion of finite product instead of notion of arbitrary direct product.

References

Pin, Jean-Éric (2016-11-30). Mathematical Foundations of Automata Theory (PDF). pp. 141–160.
Pin, Jean-Éric (1986). Varieties of formal language. New York: Plenum Publishing Corp.
Eilenberg, S (1976). Automata, languages, and machines. New york: Harcourt Brace Jovanovich Publishers. pp. chapters "Depth decomposition theorem” and “Complexity of semigroups and morphisms”.
Almeida, J (1994). Finite semigroups and universal algebra. Rivere Edge, NJ: World Scientific Publishing Co. Inc.

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