Classification of \(S_ n\)-normal semigroups
type: Item
instance of: scholarly article
title: Classification of \(S_ n\)-normal semigroups
zbMATH Open document ID: 0828.20064
DOI: 10.1007/BF02573621
author: Yanyan Li
published in: Semigroup Forum
publication date: 1996-01-09T00:00:00Z
full work available at URL: https://eudml.org/doc/135401
review text: 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 \(n < 5\).
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20B35
zbMATH DE Number: 784132
zbMATH Keywords: partial one-to-one transformations
zbMATH Keywords: symmetric groups
zbMATH Keywords: symmetric inverse semigroups
zbMATH Keywords: charts
zbMATH Keywords: transpositionals
zbMATH Keywords: alternating semigroups
zbMATH Keywords: alternating groups
zbMATH Keywords: \(S_ n\)-normal semigroups
reviewed by: jun. Kenneth D. Magill
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784132