Embedding finite involution semigroups in matrices with transposition (Q6048442)

An involution semigroup is a pair \((S,\star )\) consisting of a semigroup \(S\) with a unary operation \(\star\) that satisfies the equations \((x^{\star})^{\star}\) and \((xy)^{\star}=y^{\star}x^{\star}\) for all \(x,y\in S\). An inverse semigroup is an involution semigroup that satisfies the equations \(xx^{\star}x=x\) and \(xx^{\star}yy^{\star}=yy^{\star}xx^{\star}\) for all \(x,y\in S\). The author shows that every finite inverse semigroup is embeddable in some semigroup of binary matrices with the usual transposition \(^{T}\). Moreover, it shown that every periodic involution subsemigroup of \((\mathbb{C}^{n\times n}, ^{\ast })\), the semigroup of all \(n\times n\) complex matrices with conjugate transposition \(\ast\), is an inverse semigroup. Then the author concludes that the following conditions on any finite involution semigroup \((S,\star )\) are equivalent: \begin{itemize} \item [(a)] \((S,\star )\) is an inverse semigroup; \item [(b)] \((S,\star )\) is embeddable in some semigroup \((\mathbf{R}_{n}, ^{T})\), the semigroup of all \(n\times n\) binary matrices with at most one nonzero entry in each row and column; \item [(c)] \((S,\star )\) is embeddable in some \((\mathbb{R}^{n\times n}, ^{T} )\); \item [(d)] \((S,\star )\) is embeddable in some \((\mathbb{C}^{n\times n}, ^{\ast } )\). \end{itemize} Finally, it is shown that every finite involution semigroup is embeddable in some semigroup of binary matrices with the skew transposition.