Green index and finiteness conditions for semigroups

Robert Duncan Gray, Nik Ruskuc

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

Abstract

Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents. (c) 2008 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)3145-3164
Number of pages20
JournalJournal of Algebra
Volume320
Issue number8
Early online date12 Aug 2008
DOIs
Publication statusPublished - 15 Oct 2008

Keywords

  • Finiteness conditions
  • Index
  • Residual finiteness
  • Local finiteness
  • Complete rewriting-systems
  • Subsemigroups
  • Subgroups
  • Monoids

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