Semigroup with involution: Difference between revisions

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In [[mathematics]], particularly in [[abstract algebra]], a '''semigroup with involution''' or a '''*-semigroup''' is a [[semigroup]] equipped with an [[Involution (mathematics)|involutive]] [[anti-automorphism]], which—roughly speaking—brings it closer to a [[group (mathematics)|group]] because this involution, considered as [[unary operator]], exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application "cancelling itself out", and the same interaction law with the binary operation as in the case of the group inverse. It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.

An example from [[linear algebra]] is the [[Matrix multiplication|multiplicative]] [[monoid]] of [[Real number|real]] square [[Matrix (mathematics)|matrices]] of order ''n'' (called the [[full linear monoid]]). The [[Map (mathematics)|map]] which sends a matrix to its [[transpose]] is an involution because the transpose is well defined for any matrix and obeys the law {{nowrap|1=(''AB'')^''T'' = ''B''^''T''''A''^''T''}}, which has the same form of interaction with multiplication as taking inverses has in the [[general linear group]] (which is a subgroup of the full linear monoid). However, for an arbitrary matrix ''AA''^''T'' does not equal the identity element (namely the [[diagonal matrix]]). Another example, coming from [[formal language]] theory, is the [[free semigroup]] generated by a [[nonempty set]] (an [[Alphabet (computer science)|alphabet]]), with string [[concatenation]] as the binary operation, and the involution being the map which [[String (computer science)#Reversal|reverse]]s the [[linear order]] of the letters in a string. A third example, from basic [[set theory]], is the set of all [[binary relation]]s between a set and itself, with the involution being the [[inverse relation]], and the multiplication given by the usual [[composition of relations]].

Semigroups with involution appeared explicitly named in a 1953 paper of [[Viktor Wagner]] (in Russian) as result of his attempt to bridge the theory of semigroups with that of [[semiheap]]s.<ref name="Hollings2014">{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|year=2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=265}}</ref>