Semigroup with involution: Difference between revisions

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this should probably be some kind of history section, but there's not much else along these lines right now
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An example from [[linear algebra]] is the [[Matrix multiplication|multiplicative]] [[monoid]] of [[Real number|real]] square [[Matrix (mathematics)|matrices]] of order&nbsp;''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 (''AB'')<sup>''T''</sup> = ''B''<sup>''T''</sup>''A''<sup>''T''</sup> 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'')<sup>''T''</sup> 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>
 
==Formal definition==