Semigroups and groups

An algebraic structure or algebraic system is a nonempty set together with one or more binary operations on that set. Algebraic structures whose binary operations satisfy particularly important properties are semigroups, groups, rings, fields, modules, and so on. The simplest algebraic structure to recognize is a semigroup, which is defined as a nonempty set S with an associative binary operation. Any algebraic structure S with a binary operation + or · is normally written (S,+) or (S, ·). However, it is also customary to use an expression such as “the algebraic structure S under addition or multiplication.” Examples of semigroups are

1. (a) The systems of integers, reals, or complex numbers under usual multiplication (or addition)

2. (b) The set of mappings from a nonempty set S into itself under composition of mappings

3. (c) The set of n × n matrices over complex numbers under multiplication (or addition) of matrices

Let (S,·) be a semigroup and let a,b ∈ S. We usually write ab instead of a · b. An element e in S is called a left identity if ea = a for all a ∈ S. A right identity is defined similarly. It is possible to have a semigroup with several left identities or several right identities. However, if a semigroup S has both a left identity e and a right identity f, then e = ef = f. Therefore, e is the unique two-sided identity of the semigroup.