Semigroup with involution: Difference between revisions

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A semigroup ''S'' with an involution * is called a '''*-regular semigroup''' (in the sense of Drazin) if for every ''x'' in ''S'', ''x''* is ''H''-equivalent to some inverse of ''x'', where ''H'' is the [[Green's relations|Green’s relation]] ''H''. This defining property can be formulated in several equivalent ways. Another is to say that every [[Green's relations#The L.2C R.2C and J relations|''L''-class]] contains a projection. An axiomatic definition is the condition that for every ''x'' in ''S'' there exists an element ''x′x''′ such that {{nowrap|1=''x′xx′x''′''xx′'' = ''x′x''′}}, {{nowrap|1=''xx′xxx''′''x'' = ''x''}}, {{nowrap|1=(''xx′xx''′)* = ''xx′xx''′}}, {{nowrap|1=(''x′xx''′''x'')* = ''x′xx''′''x''}}. [[Michael P. Drazin]] first proved that given ''x'', the element ''x′'' satisfying these axioms is unique. It is called the Moore–Penrose inverse of ''x''. This agrees with the classical definition of the [[Moore–Penrose inverse]] of a square matrix.

In the [[Matrix multiplication|multiplicative]] semigroup ''M''_''n'' ( ''C'' ) of square matrices of order ''n'', the map which assigns a matrix ''A'' to its [[Hermitian conjugate]] ''A''* is an involution. The semigroup ''M''_''n'' ( ''C'' ) is a *-regular semigroup with this involution. The Moore–Penrose inverse of A in this *-regular semigroup is the classical Moore–Penrose inverse of ''A''.

==Free semigroup with involution ==