Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): x ⋅ y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x, y).
Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses.
A monoid is an algebraic structure intermediate between semigroups and groups, and is a semigroup having an identity element, thus obeying all but one of the axioms of a group: existence of inverses is not required of a monoid.
Some other techniques for studying semigroups, like Green's relations, do not resemble anything in group theory.
[1] In other areas of applied mathematics, semigroups are fundamental models for linear time-invariant systems.
There are also interesting classes of semigroups that do not contain any groups except the trivial group; examples of the latter kind are bands and their commutative subclass – semilattices, which are also ordered algebraic structures.
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}.
[3] Similarly, every magma has at most one absorbing element, which in semigroup theory is called a zero.
An example of a semigroup with no minimal ideal is the set of positive integers under addition.
Congruence classes and factor monoids are the objects of study in string rewriting systems.
Here the term maximal subgroup differs from its standard use in group theory.
For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent.
[10] 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 that satisfies the additional idempotence law a ∧ a = a.
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.
[11] There is an obvious semigroup homomorphism j : S → G(S) that sends each element of S to the corresponding generator.
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.
[16] Semigroup theory can be used to study some problems in the field of partial differential equations.
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 Lp space of square-integrable real-valued functions with domain the interval (0, 1) and let A be the second-derivative operator with domain where
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.
The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as groups or rings.
The term is used in English in 1908 in Harold Hinton's Theory of Groups of Finite Order.
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.
[18] From that point on, the foundations of semigroup theory were further laid by David Rees, James Alexander Green, Evgenii Sergeevich Lyapin [fr], Alfred H. Clifford and Gordon Preston.
[19] At an algebraic conference in 1972 Schein surveyed the literature on BA, the semigroup of relations on A.
[21] 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.
n-ary associativity is a string of length n + (n − 1) with any n adjacent elements bracketed.
A third generalization is the semigroupoid, in which the requirement that the binary relation be total is lifted.
