On semigroups with involution

A semigroup S with an involution * is called a special involution semigroup if and only if, for every finite nonempty subset T of S , . It is shown that a semigroup is inverse if and only if it is a special involution semigroup in which every element invariant under the involution is periodic. Other examples of special involution semigroups are discussed; these include free semigroups, totally ordered cancellative commutative semigroups and certain semigroups of matrices. Some properties of the semigroup algebras of special involution semigroups are also derived. In particular, it is shown that their real and complex semigroup algebras are semiprimitive.