About: Semigroup with involution

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In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a 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.

Attributes	Values
rdfs:label	Semigroup with involution (en)
rdfs:comment	In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a 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. (en)
sameAs	Semigroup with involution Semigroup with involution
dbp:wikiPageUsesTemplate	dbt:Anchor dbt:Expand_section dbt:Reflist dbt:Tmath dbt:Use_dmy_dates dbt:Isbn dbt:PlanetMath_attribution
Subject	Algebraic structures Semigroup theory
gold:hypernym	Semigroup
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Semigroup_with_involution?oldid=1003639006&ns=0
Wikipage page ID	22679231 (xsd:integer)
page length (characters) of wiki page	25709 (xsd:nonNegativeInteger)
Wikipage revision ID	1003639006 (xsd:integer)
has abstract	In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a 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 multiplicative monoid of real square matrices of order n (called the full linear monoid). The 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)T = BTAT, 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, AAT 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), with string concatenation as the binary operation, and the involution being the map which reverses the linear order of the letters in a string. A third example, from basic set theory, is the set of all binary relations between a set and itself, with the involution being the converse 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 semiheaps. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Semigroup_with_involution
is Wikipage redirect of	Baer -semigroup Free half group Shamir congruence -regular semigroup *-semigroup Monoid with involution Dyck congruence Foulis semigroup Involutive monoid Free monoid with involution Free semigroup with involution
is Link from a Wikipage to another Wikipage of	Baer -semigroup -algebra Antihomomorphism Baer ring Binary relation Moore–Penrose inverse C-algebra David J. Foulis Distributive property Dagger category Composition of relations Converse relation Catalan number Homogeneous relation Michael P. Drazin Free half group Inverse element Inverse semigroup Involution (mathematics) -regular semigroup *-semigroup Monoid with involution Dyck congruence Foulis semigroup Involutive monoid Free monoid with involution Free semigroup with involution

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