Classification of \(S_ n\)-normal semigroups
Let \(S_n\) denote the symmetric group of permutations on \(N=\{1,2,\dots,n\}\) and let \(C_n\) denote the symmetric inverse semigroup of charts (partial one-to-one transformations on \(N\)). A subsemigroup \(T\) of \(C_n\) is \(S_n\)-normal if \(\alpha^{-1} T\alpha\subseteq T\) for all \(\alpha\in S_n\). Denote by \(I_r\) the ideal of \(C_n\) of all charts of rank \(\leq r\) and let \(E_r=\{e\in C_n:e^2=e\) and \(\text{rank }e\leq r\}\). A transpositional is defined to be a chart that is either a transposition \((ij)\) or a semitransposition \((ij]\) (the closed bracket on the right indicates that \(j\) does not belong to the domain of the chart) and the subsemigroup of \(C_n\) consisting of all those charts which are products of an even number of transpositionals is referred to as the alternating semigroup and is denoted by \(A^c_n\). The usual alternating group is denoted by \(A_n\). Finally, let \(\delta\) denote the identity map on \(N\). The authors show that if \(n\geq 5\), then a subsemigroup \(T\) of \(C_n\) is \(S_n\)-normal if and only if \(T=S\) or \(T=S\cup\{\delta\}\) where \(S\) is one of the following semigroups: (1) a normal subgroup \(G\) of \(S_n\), (2) \(G\cup I_r\), \(0\leq r\leq n -1\) where \(G\) is a normal subgroup of \(S_n\), (3) the alternating semigroup \(A^c_n\), (4) the ideal \(A^c_n\setminus A_n\) of the alternating semigroup \(A^c_n\), (5) a union \(E_m\cup I_r\), \(0\leq r\leq m\leq n-1\), (6) a union \(E_m\cup U\cup I_{r-1}\), \(2\leq r\leq m\leq m-1\) where \(U=\bigvee_{1\leq i\leq{n\choose r}} G_i\) is a disjoint union of nontrivial conjugate groups, (7) a union \(E_m\cup X_n\), \(n/2\leq m\leq n-1\) for even \(n\) where \(X_n\) is the subsemigroup of \(C_n\) generated by all the charts which are conjugate to a particular chart. The authors then conclude by also classifying the \(S_n\)-normal subsemigroups of \(C_n\) for \(n < 5\).
scientific article; zbMATH DE number 784132