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==Free semigroup with involution ==
As with all varieties, the [[category theory|category]] of semigroups with involution admits [[free object]]s. The construction of a free semigroup (or monoid) with involution is based on that of a [[free semigroup]] (and respectively that of a free monoid). Moreover, the construction of a [[free group]] can easily be derived by refining the construction of a free monoid with involution.<ref name="L51">{{harvcoltxt|Lawson
The [[Generator (mathematics)|generators]] of a free semigroup with involution are the elements of the union of two ([[equinumerous]]) [[disjoint sets]] in [[Bijection|bijective correspondence]]: <math>Y=X\sqcup X^\dagger</math>. (Here the notation <math>\sqcup\,</math> emphasized that the union is actually a [[disjoint union]].) In the case were the two sets are finite, their union ''Y'' is sometimes called an ''[[Alphabet (computer science)|alphabet]] with involution''<ref name="EhrenfeuchtHarju1999">{{cite book|author1=Andrzej Ehrenfeucht|author2=T. Harju|author3=Grzegorz Rozenberg|title=The Theory of 2-structures: A Framework for Decomposition and Transformation of Graphs|year=1999|publisher=World Scientific|isbn=978-981-02-4042-4|pages=13–14}}</ref> or a ''symmetric alphabet''.<ref name="Sakarovitch">{{cite book|title=Elements of Automata Theory|publisher=Cambridge University Press|pages=305–306|author=Jacques Sakarovitch}}</ref> Let <math>\theta:X\rightarrow X^\dagger</math> be a bijection; <math>\theta</math> is naturally [[Function (mathematics)#Restrictions and extensions|extended]] to a bijection <math>{ }\dagger: Y \to Y</math> essentially by taking the disjoint union of <math>\theta</math> (as a set) with its [[inverse function|inverse]], or in [[piecewise]] notation:<ref name="Lipscomb1996">{{cite book|author=Stephen Lipscomb|title=Symmetric Inverse Semigroups|year=1996|publisher=American Mathematical Soc.|isbn=978-0-8218-0627-2|page=86}}</ref>
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