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#If ''S'' is an [[inverse semigroup]] then the inversion map is an involution which leaves the [[idempotent]]s [[Invariant (mathematics)|invariant]]. As noted in the previous example, the inversion map is not necessarily the only map with this property in an inverse semigroup; there may well be other involutions that leave all idempotents invariant; for example the identity map on a commutative regular, hence inverse, semigroup, in particular, an abelian group. A [[regular semigroup]] is an [[inverse semigroup]] if and only if it admits an involution under which each idempotent is an invariant.<ref>Munn, Lemma 1</ref>
#Underlying every [[C*-algebra]] is a *-semigroup. An important [[C*-algebra#Finite-dimensional C*-algebras|instance]] is the algebra ''M''<sub>''n''</sub>('''C''') of ''n''-by-''n'' [[matrix (mathematics)|matrices]] over '''[[Complex number|C]]''', with the [[conjugate transpose]] as involution.
# {{anchor|ex5}} If ''X'' is a set, the set of all [[binary relation]]s on ''X'' is a *-semigroup with the * given by the [[inverse relation]], and the multiplication given by the usual [[composition of relations]]. This is an example of a *-semigroup which is not a regular semigroup.
# {{anchor|ex6}} If X is a set, then the set of all finite sequences (or [[String (computer science)|strings]]) of members of X forms a [[free monoid]] under the operation of concatenation of sequences, with sequence reversal as an involution.
# {{anchor|ex7}} A [[rectangular band]] on a Cartesian product of a set ''A'' with itself, i.e. with elements from ''A'' × ''A'', with the semigroup product defined as (''a'', ''b'')(''c'', ''d'') = (''a'', ''d''), with the involution being the order reversal of the elements of a pair (''a'', ''b'')* = (''b'', ''a''). This semigroup is also a [[regular semigroup]], as all bands are.<ref>Nordahl and Scheiblich</ref>
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