Semigroup with involution: Difference between revisions

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The construction above is actually the only way to extend a given map <math>\theta\,</math> from <math>X\,</math> to <math>X^\dagger\,</math>, to an involution on <math>Y^+\,</math> (and likewise on <math>Y^*\,</math>). The qualifier "free" for these constructions is justified in the usual sense that they are [[universal algebra|universal construction]]s. In the case of the free semigroup with involution, given an arbitrary semigroup with involution <math>S\,</math> and a map <math>\Phi:X\rightarrow S</math>, then a [[semigroup homomorphism]] <math>\overline\Phi:(X\sqcup X^\dagger)^+\rightarrow S</math> exists such that <math>\Phi = \iota \circ \overline\Phi</math>, where <math>\iota : X \rightarrow (X\sqcup X^\dagger)^+</math> is the [[inclusion map]] and [[composition of functions]] is taken in [[Function composition#Alternative notations|diagram order]].<ref name="L172"/> The construction of <math>(X\sqcup X^\dagger)^+</math> as a semigroup with involution is unique up to [[isomorphism]]. An analogous argument holds for the free monoid with involution in terms of [[monoid homomorphism]]s and the uniqueness up to isomorphism of the construction of <math>(X\sqcup X^\dagger)^*</math> as a monoid with involution.

The construction of a [[free group]] is not very far off from that of a free monoid with involution. The additional ingredient needed is to define a notion of [[reduced word]] and a [[rewriting]] rule for producing such words simply by deleting any adjacent pairs of letter of the form <math>xx^\dagger</math> or <math>x^\dagger x</math>. It can be shown than the order of rewriting (deleting) such pairs does not matter, i.e. any order of deletions produces the same result.<ref name="L51"/> (Otherwise put, these rules define a [[Confluence (abstract rewriting)|confluent]] rewriting system.) Equivalently, a free group is constructed from a free monoid with involution by taking the [[Quotientquotient (universal algebra)|quotient]] of the latter by the [[Congruence relation|congruence]] <math>\{ (yy^\dagger, \varepsilon) : y\in Y\}</math>, which is sometimes called the '''Dyck congruence'''&mdash;in a certain sense it generalizes [[Dyck language]] to multiple kinds of "parentheses" However simplification in the Dyck congruence takes place regardless of order. For example, if ")" is the inverse of "(", then <math>()=)(=\varepsilon</math>; the one-sided congruence that appears in the Dyck language proper <math>\{ (xx^\dagger, \varepsilon) : x\in X\}</math>, which instantiates only to <math>()=\varepsilon</math> is (perhaps confusingly) called the '''Shamir congruence'''. The quotient of a free monoid with involution by the Shamir congruence is not a group, but a monoid <!--with involution?-->; nevertheless it has been called the '''free half group''' by its first discoverer&mdash;[[Eli Shamir]]&mdash;although more recently it has been called the '''involutive monoid''' generated by ''X''.<ref name="Sakarovitch"/><ref name="DrosteKuich2009">{{cite book|editors=Manfred Droste, Werner Kuich, Heiko Vogler|title=Handbook of Weighted Automata|year=2009|publisher=Springer |isbn=978-3-642-01492-5|page=271|authors=Ion Petre and [[Arto Salomaa]]|chapter=Algebraic Systems and Pushdown Automata}}</ref> (This latter choice of terminology conflicts however with the use of "involutive" to denote any semigroup with involution&mdash;a practice also encountered in the literature.<ref name="Neeb2000">{{cite book|author=Karl-Hermann Neeb|title=Holomorphy and Convexity in Lie Theory|year=2000|publisher=Walter de Gruyter|isbn=978-3-11-015669-0|page=21}}</ref><ref name="BeltramettiCassinelli2010">{{cite book|author1=Enrico G. Beltrametti|author2=Gianni Cassinelli|title=The Logic of Quantum Mechanics|year=2010|origyear=1981|publisher=Cambridge University Press|isbn=978-0-521-16849-6|page=178}}</ref>)

== Baer *-semigroups ==