Green’s relations and regularity in centralizers of permutations. (English) Zbl 0924.20049

Konieczny, Janusz

Glasg. Math. J. 41, No. 1, 45-57 (1999).

The author continues his study of the centralizer, \(C(\sigma)\), of a partial transformation \(\sigma\in PT_n\). In this paper \(\sigma\) is a permutation of the base set and Green’s relations and criteria for regularity of \(C(\sigma)\) are found. The characterization of the Green’s class for \(\alpha\in C(\sigma)\) involves the partial transformation of the cycle set of \(\sigma\) induced by \(\alpha\) and the cycle length \(\ell(a)\), \(\ell(b)\) etc. of the cycles \(a\), \(b\) etc. of \(\sigma\).
For example, \(C(\sigma)\) is regular if and only if for all cycles \(a,b\in C(\sigma)\), \(\ell(b)\) divides \(\ell(a)\) implies \(\ell(b)=\ell(a)\). If we strengthen the right hand side of this implication to \(b=a\), then we have a characterization for \(C(\sigma)\) to be an inverse semigroup (which must be a semilattice of groups).

Reviewer: P.M.Higgins (Colchester)

MSC:

20M20

Semigroups of transformations, relations, partitions, etc.

Keywords:

centralizers; partial transformations; permutations; Green’s relations; regularity; cycles; inverse semigroups; semilattices of groups