Semigroups

Abbreviation: Sgrp

Definition

A semigroup is a structure S = (S,·), where · is an infix binary operation, called the semigroup product, such that
· is associative:   (xy)z = x(yz).

Morphisms

Let S and T be semigroups. A morphism from S to T is a function h : S→T that is a homomorphism: h(xy) = h(x)h(y).

Some results

Examples

(X^X,o), the collection of functions on a sets X, with composition.

(Σ ⁺ ,·), the collection of nonempty strings over Σ, with concatenation.

Properties

Finite members

Size 1:  1
Size 2:  5
Size 3:  24
Size 4:  188
[Size 5]?:  1915
[Size 6]?:  28634
[Size 7]?:  1627672
[Semigroups in the Encyclopedia of Integer Sequences]

Subclasses

Superclasses