Semigroup

Definition

A semigroup is a set S together with a binary operation " ⋅ " (that is, a function ⋅ : S × S → S ) that satisfies the associative property:

More succinctly, a semigroup is an associative magma.

Examples of semigroups

Identity and zero

A left identity of a semigroup S (or more generally, magma) is an element e such that for all x in S , e x = x . Similarly, a right identity is an element f such that for all x in S , x f = x . Left and right identities are both called one-sided identities. A semigroup may have one or more left identities but no right identity, and vice versa.

A two-sided identity (or just identity) is an element that is both a left and right identity. Semigroups with a two-sided identity are called monoids. A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup. If a semigroup has both a left identity and a right identity, then it has a two-sided identity (which is therefore the unique one-sided identity).

A semigroup S without identity may be embedded in a monoid formed by adjoining an element e ∉ S to S and defining e ⋅ s = s ⋅ e = s for all s ∈ S ∪ { e } . The notation S 1 denotes a monoid obtained from S by adjoining an identity if necessary ( S 1 = S for a monoid).

Similarly, every magma has at most one absorbing element, which in semigroup theory is called a zero. Analogous to the above construction, for every semigroup S , one can define S 0 , a semigroup with 0 that embeds S .

Subsemigroups and ideals

The semigroup operation induces an operation on the collection of its subsets: given subsets A and B of a semigroup S, their product A · B, written commonly as AB, is the set { ab | a in A and b in B }. (This notion is defined identically as it is for groups.) In terms of this operation, a subset A is called

If A is both a left ideal and a right ideal then it is called an ideal (or a two-sided ideal).

If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S. So the subsemigroups of S form a complete lattice.

An example of semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal of a commutative semigroup, when it exists, is a group.

Green's relations, a set of five equivalence relations that characterise the elements in terms of the principal ideals they generate, are important tools for analysing the ideals of a semigroup and related notions of structure.

The subset with the property that its every element commutes with any other element of the semigroup is called the center of the semigroup. The center of a semigroup is actually a subsemigroup.

Homomorphisms and congruences

A semigroup homomorphism is a function that preserves semigroup structure. A function f: S → T between two semigroups is a homomorphism if the equation

holds for all elements a, b in S, i.e. the result is the same when performing the semigroup operation after or before applying the map f.

A semigroup homomorphism between monoids preserves identity if it is a monoid homomorphism. But there are semigroup homomorphisms which are not monoid homomorphisms, e.g. the canonical embedding of a semigroup S without identity into S 1 . Conditions characterizing monoid homomorphisms are discussed further. Let f : S 0 → S 1 be a semigroup homomorphism. The image of f is also a semigroup. If S 0 is a monoid with an identity element e 0 , then f ( e 0 ) is the identity element in the image of f . If S 1 is also a monoid with an identity element e 1 and e 1 belongs to the image of f , then f ( e 0 ) = e 1 , i.e. f is a monoid homomorphism. Particularly, if f is surjective, then it is a monoid homomorphism.

Two semigroups S and T are said to be isomorphic if there is a bijection f : S ↔ T with the property that, for any elements a, b in S, f(ab) = f(a)f(b). Isomorphic semigroups have the same structure.

A semigroup congruence ∼ is an equivalence relation that is compatible with the semigroup operation. That is, a subset ∼ ⊆ S × S that is an equivalence relation and x ∼ y and u ∼ v implies x u ∼ y v for every x , y , u , v in S. Like any equivalence relation, a semigroup congruence ∼ induces congruence classes

and the semigroup operation induces a binary operation ∘ on the congruence classes:

Because ∼ is a congruence, the set of all congruence classes of ∼ forms a semigroup with ∘ , called the quotient semigroup or factor semigroup, and denoted S / ∼ . The mapping x ↦ [ x ] ∼ is a semigroup homomorphism, called the quotient map, canonical surjection or projection; if S is a monoid then quotient semigroup is a monoid with identity [ 1 ] ∼ . Conversely, the kernel of any semigroup homomorphism is a semigroup congruence. These results are nothing more than a particularization of the first isomorphism theorem in universal algebra. Congruence classes and factor monoids are the objects of study in string rewriting systems.

A nuclear congruence on S is one which is the kernel of an endomorphism of S.

A semigroup S satisfies the maximal condition on congruences if any family of congruences on S, ordered by inclusion, has a maximal element. By Zorn's lemma, this is equivalent to saying that the ascending chain condition holds: there is no infinite strictly ascending chain of congruences on S.

Every ideal I of a semigroup induces a subsemigroup, the Rees factor semigroup via the congruence x ρ y   ⇔   either x = y or both x and y are in I.

Structure of semigroups

For any subset A of S there is a smallest subsemigroup T of S which contains A, and we say that A generates T. A single element x of S generates the subsemigroup { xⁿ | n is a positive integer }. If this is finite, then x is said to be of finite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finite order. A semigroup generated by a single element is said to be monogenic (or cyclic). If a monogenic semigroup is infinite then it is isomorphic to the semigroup of positive integers with the operation of addition. If it is finite and nonempty, then it must contain at least one idempotent. It follows that every nonempty periodic semigroup has at least one idempotent.

A subsemigroup which is also a group is called a subgroup. There is a close relationship between the subgroups of a semigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of the subgroup. For each idempotent e of the semigroup there is a unique maximal subgroup containing e. Each maximal subgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups. Here the term maximal subgroup differs from its standard use in group theory.

More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent. The number of finite semigroups of a given size (greater than 1) is (obviously) larger than the number of groups of the same size. For example, of the sixteen possible "multiplication tables" for a set of two elements {a, b}, eight form semigroups whereas only four of these are monoids and only two form groups. For more on the structure of finite semigroups, see Krohn–Rhodes theory.

Special classes of semigroups

Structure theorem for commutative semigroups

There is a structure theorem for commutative semigroups in terms of semilattices. A semilattice (or more precisely a meet-semilattice) ( L , ≤ ) is a partially ordered set where every pair of elements a , b ∈ L has a greatest lower bound, denoted a ∧ b . The operation ∧ makes L into a semigroup satisfying the additional idempotence law a ∧ a = a .

Given a homomorphism f : S → L from an arbitrary semigroup to a semilattice, each inverse image S a = f − 1 { a } is a (possibly empty) semigroup. Moreover, S becomes graded by L , in the sense that

S a S b ⊆ S a ∧ b

If f is onto, the semilattice L is isomorphic to the quotient of S by the equivalence relation ∼ such that x ∼ y iff f ( x ) = f ( y ) . This equivalence relation is a semigroup congruence, as defined above.

Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup. The structure theorem says that for any commutative semigroup S , there is a finest congruence ∼ such that the quotient of S by this equivalence relation is a semilattice. Denoting this semilattice by L , we get a homomorphism f from S onto L . As mentioned, S becomes graded by this semilattice.

Furthermore, the components S a are all Archimedean semigroups. An Archimedean semigroup is one where given any pair of elements x , y , there exists an element z and n > 0 such that x n = y z .

The Archimedean property follows immediately from the ordering in the semilattice L , since with this ordering we have f ( x ) ≤ f ( y ) if and only if x n = y z for some z and n > 0 .

Group of fractions

The group of fractions or group completion of a semigroup S is the group G = G(S) generated by the elements of S as generators and all equations xy = z which hold true in S as relations. There is an obvious semigroup homomorphism j : S → G(S) which sends each element of S to the corresponding generator. This has a universal property for morphisms from S to a group: given any group H and any semigroup homomorphism k : S → H, there exists a unique group homomorphism f : G → H with k=fj. We may think of G as the "most general" group that contains a homomorphic image of S.

An important question is to characterize those semigroups for which this map is an embedding. This need not always be the case: for example, take S to be the semigroup of subsets of some set X with set-theoretic intersection as the binary operation (this is an example of a semilattice). Since A.A = A holds for all elements of S, this must be true for all generators of G(S) as well: which is therefore the trivial group. It is clearly necessary for embeddability that S have the cancellation property. When S is commutative this condition is also sufficient and the Grothendieck group of the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroups can be traced to the first substantial paper on semigroups. Anatoly Maltsev gave necessary and sufficient conditions for embeddability in 1937.

Semigroup methods in partial differential equations

Semigroup theory can be used to study some problems in the field of partial differential equations. Roughly speaking, the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equation on a function space. For example, consider the following initial/boundary value problem for the heat equation on the spatial interval (0, 1) ⊂ R and times t ≥ 0:

Let X = L²((0, 1) R) be the L^p space of square-integrable real-valued functions with domain the interval (0, 1) and let A be the second-derivative operator with domain

where H² is a Hardy space. Then the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space X:

On an heuristic level, the solution to this problem "ought" to be u(t) = exp(tA)u₀. However, for a rigorous treatment, a meaning must be given to the exponential of tA. As a function of t, exp(tA) is a semigroup of operators from X to itself, taking the initial state u₀ at time t = 0 to the state u(t) = exp(tA)u₀ at time t. The operator A is said to be the infinitesimal generator of the semigroup.

History

The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as groups or rings. A number of sources attribute the first use of the term (in French) to J.-A. de Séguier in Élements de la Théorie des Groupes Abstraits (Elements of the Theory of Abstract Groups) in 1904. The term is used in English in 1908 in Harold Hinton's Theory of Groups of Finite Order.

Anton Suschkewitsch obtained the first non-trivial results about semigroups. His 1928 paper Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit (On finite groups without the rule of unique invertibility) determined the structure of finite simple semigroups and showed that the minimal ideal (or Green's relations J-class) of a finite semigroup is simple. From that point on, the foundations of semigroup theory were further laid by David Rees, James Alexander Green, Evgenii Sergeevich Lyapin, Alfred H. Clifford and Gordon Preston. The latter two published a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodical called Semigroup Forum (currently edited by Springer Verlag) became one of the few mathematical journals devoted entirely to semigroup theory.

In recent years researchers in the field have become more specialized with dedicated monographs appearing on important classes of semigroups, like inverse semigroups, as well as monographs focusing on applications in algebraic automata theory, particularly for finite automata, and also in functional analysis.

Generalizations

If the associativity axiom of a semigroup is dropped, the result is a magma, which is nothing more than a set M equipped with a binary operation M × M → M.

Generalizing in a different direction, an n-ary semigroup (also n-semigroup, polyadic semigroup or multiary semigroup) is a generalization of a semigroup to a set G with a n-ary operation instead of a binary operation. The associative law is generalized as follows: ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. N-ary associativity is a string of length n + (n − 1) with any n adjacent elements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to an n-ary group.

A third generalization is the semigroupoid, in which the requirement that the binary relation be total is lifted. As categories generalize monoids in the same way, a semigroupoid behaves much like a category but lacks identities.

Infinitary generalizations of commutative semigroups have sometimes been considered by various authors.