# Involution semigroup

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 20M10 [MSN][ZBL]

A semigroup $(S,{\cdot})$ with an involution $*$, having the properties $(x\cdot y)^* = y^* \cdot x^*$ and $x^{{*}{*}} = x$.

A *projection* in an involution semigroup is an element $e$ such that $e\cdot e = e = e^*$. There is a partial order on projections given by $e \le f$ if $e\cdot f = e$.

A *regular involution* semigroup is one in which $x x^* x = x$: that is, it is also a regular semigroup in which the involution acts as generalised inverse.

#### References

- Ivan Rival (ed.),"Algorithms and Order", Kluwer (1989) ISBN 940107691X Zbl 0709.68004

**How to Cite This Entry:**

Involution semigroup.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Involution_semigroup&oldid=51369