Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-12-05T02:29:19.754Z Has data issue: false hasContentIssue false
  • English
  • Français

Commutative orders

Published online by Cambridge University Press:  14 November 2011

David Easdown  and
Victoria Gould
David Easdown
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia
Victoria Gould
Affiliation:
Department of Mathematics, University of York, Heslington, York YOl 5DD, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

A subsemigroup S of a semigroup Q is a left (right) order in Q if every q ∈ Q can be written as q = a*b(q = ba*) for some a, b ∈S, where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. If S is both a left order and a right order in Q, we say that S is an order in Q. We show that if S is a left order in Q and S satisfies a permutation identity xl…xn = x1π…xnπ where 1 < 1π and nπ<n, then S and Q are commutative. We give a characterisation of commutative orders and decide the question of when one semigroup of quotients of a commutative semigroup is a homomorphic image of another. This enables us to show that certain semigroups have maximum and minimum semigroups of quotients. We give examples to show that this is not true in general.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 6 , 1996 , pp. 1201 - 1216
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Clifford, A. H. and Preston, G. B.. The Algebraic Theory of Semigroups, vol. I (Providence, RI: American Mathematical Society, 1961).Google Scholar
2Fountain, J. B.. Abundant semigroups. Proc. London Math. Soc. 44 (1982, 103–29.CrossRefGoogle Scholar
3Fountain, J. B. and Petrich, M.. Completely 0-simple semigroups of quotients. J. Algebra 101 (1986), 365–402.CrossRefGoogle Scholar
4Gould, V. A. R.. Clifford semigroups of left quotients. Glasgow Math. J. 28 (1986), 181–91.CrossRefGoogle Scholar
5Gould, V. A. R.. Straight left orders. Studia. Set Math. Hungar. (to appear).Google Scholar
6Howie, J. M.. An Introduction to Semigroup Theory (New York: Academic Press, 1976).Google Scholar
7Khan, N. M.. On saturated permutative varieties and consequences of permutation identities. J. Austral. Math. Soc. 38 (1985), 186–97.CrossRefGoogle Scholar
8McAlister, D. B.. One-to-one partial right translations of a right cancellative semigroup. J. Algebra 43 (1976), 231–51.CrossRefGoogle Scholar
9Smith, P. M.. Orders in completely regular semigroups (D.Phil. Thesis, University of York, 1990).Google Scholar
10Yamada, M.. Regular semigroups whose idempotents satisfy permutation identities. Pacific J. Math. 21 (1967), 371–92.CrossRefGoogle Scholar