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Semigroup with involution
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly in <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a semigroup with involution or a *-semigroup is a <a href="/facts/Semigroup/rgSdk2kt">semigroup</a> equipped with an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involutive</a> <a href="/facts/Anti-automorphism/nIRWxknY">anti-automorphism</a>, which—roughly speaking—brings it closer to a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> because this involution, considered as <a href="/facts/Unary_operator/kb6CkgsX">unary operator</a>, exhibits certain fundamental properties of the operation of taking the inverse in a group: </p> <ul><li><a href="/facts/Uniqueness_(mathematics)/qBpdIqel">Uniqueness</a></li> <li>Double application "cancelling itself out".</li> <li>The same interaction law with the binary operation as in the case of the group inverse.</li></ul> <p>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. </p><p>An example from <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> is a set of real-valued n-by-n square matrices with the matrix-transpose as the involution. The <a href="/facts/Map_(mathematics)/FABF1l2f">map</a> which sends a matrix to its <a href="/facts/Transpose/8wmsagGS">transpose</a> is an involution because the transpose is well defined for any matrix and obeys the law (<i>AB</i>)T = <i>B</i>T<i>A</i>T, which has the same form of interaction with multiplication as taking inverses has in the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> (which is a subgroup of the full linear monoid). However, for an arbitrary matrix, <i>AA</i>T does not equal the identity element (namely the <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a>). Another example, coming from <a href="/facts/Formal_language/crDTyP8q">formal language</a> theory, is the <a href="/facts/Free_semigroup/h3nPVjqr">free semigroup</a> generated by a <a href="/facts/Nonempty_set/ffEk3eA6">nonempty set</a> (an <a href="/facts/Alphabet_(computer_science)/6KW9qYRW">alphabet</a>), with string <a href="/facts/Concatenation/1npYg6rc">concatenation</a> as the binary operation, and the involution being the map which <a href="/facts/String_(computer_science)/4ChJu5WS">reverses</a> the <a href="/facts/Linear_order/xnTtmuYJ">linear order</a> of the letters in a string. A third example, from basic <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, is the set of all <a href="/facts/Binary_relation/r9BwGgaK">binary relations</a> between a set and itself, with the involution being the <a href="/facts/Converse_relation/EVGlRSz7">converse relation</a>, and the multiplication given by the usual <a href="/facts/Composition_of_relations/tD8joBlo">composition of relations</a>. </p><p>Semigroups with involution appeared explicitly named in a 1953 paper of <a href="/facts/Viktor_Wagner/CoKBxcNe">Viktor Wagner</a> (in Russian) as result of his attempt to bridge the theory of semigroups with that of <a href="/facts/Semiheap/9Say1QvS">semiheaps</a>. </p>