Semigroup with involution

In mathematics, in semigroup theory, an involution in a semigroup is a transformation of the semigroup which is its own inverse and which is an anti-automorphism of the semigroup. A semigroup in which an involution is defined is called a semigroup with involution. In the multiplicative semigroup of real square matrices of order n, the map which sends a matrix to its transpose is an involution. In the free semigroup generated by a nonempty set the operation which reverses the order of the letters in a word is an involution. A semigroup with an involution is also called a *– semigruop.

Formal definition

Let S be a semigroup. An involution in S is a unary operation * on S (or, a transformation * : S → S, x → x*) satisfying the following conditions:

1. For all x in S, (x*)* = x.
2. For all x, y in S we have ( xy )* = y*x*.

The semigroup S with the involution * is called a semigroup with involution.

Examples

1. If S is a commutative semigroup then the identity map of S is an involution.
2. If S is a group then the inversion map * : S → S defined by x* = x⁻¹ is an involution.
3. If S is an inverse semigroup then the inversion map is an involution which leaves the idempotent invariant. Interestingly the converse is also true. A semigroup is an inverse semigroup if and only if it admits an involution under which each idempotent is an invariant.

* – regular semigroups

A semigroup S with an ivolution * is called a * – regular semigroup if for every x in S, x* is H-equivalent to some inverse of x, where H is the Green’s relation H. This defining property can be formulated in several equivalent ways. For example, it is equivalent to the condition that for every x in S there exists an element x’ such that x’xx’ = x’, xx’x = x, ( xx’ )* = xx’, ( x’x )* = x’x. Given x the element x’ is unique. It is called the Moore–Penrose inverse of x. This agrees with the classical definition of the Moore-Penrose inverse of a square matrix. In the multiplicative semigroup M_n ( C ) of square matrices of order n, the map which assigns a matrix A to its Hermitian conjugate A* is an involution. The semigroup M_n ( C ) is a * – regular semigroup with this involution. The Moore–Penrose inverse of A in this * – regular semigroup is the classical Moore–Penrose inverse of A.

See also

Special classes of semigroups

References

• Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". World Scientific ISBN 9810233167
• D J Foulis (1958). Involution Semigroups, Ph.D. Thesis, Tulane University, New Orleans, LA. Publications of D.J. Foulis (Accessed on 05 May 2009)