Hostname: page-component-7bb8b95d7b-2h6rp Total loading time: 0 Render date: 2024-09-09T18:52:40.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Independence algebras, basis algebras and semigroups of quotients

Part of: Semigroups Algebraic structures

Published online by Cambridge University Press:  05 August 2010

Victoria Gould
Victoria Gould
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK (varg1@york.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if A is a stable basis algebra satisfying the distributivity condition, then B is a reduct of an independence algebra A having the same rank. If this rank is finite, then the endomorphism monoid of B is a left order in the endomorphism monoid of A.

Keywords

semigroupindependencebasisexchange propertyendomorphism monoidquotients

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 08A35: Automorphisms, endomorphisms 20M10: General structure theory 08A05: Structure theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 3 , October 2010 , pp. 697 - 729
Copyright
Copyright © Edinburgh Mathematical Society 2010

References

1. Araújo, J., Idempotent generated endomorphisms of an independence algebra, Semigroup Forum 132(2003), 464–467.CrossRefGoogle Scholar
2. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Mathematical Surveys, Volume 1 (American Mathematical Society, Providence, RI, 1961).Google Scholar
3. Cohn, P. M., Universal algebra(Harper & Row, New York, 1965).Google Scholar
4. Erdös, J. A., On products of idempotent matrices, Glasgow Math. J. 132(1967), 118–122.CrossRefGoogle Scholar
5. Fountain, J., Products of idempotent integer matrices, Math. Proc. Camb. Phil. Soc. 132(1991), 431–441.CrossRefGoogle Scholar
6. Fountain, J. and Gould, V., Relatively free algebras with weak exchange properties, J. Austral. Math. Soc. 132(2003), 355–384 (see also http://www-users.york.ac.uk/˜varg1).CrossRefGoogle Scholar
7. Fountain, J. and Gould, V., Endomorphisms of relatively free algebras with weak exchange properties, Alg. Univers. 132(2004), 257–285 (see also http://www-users.york.ac.uk/˜varg1).Google Scholar
8. Fountain, J. and Gould, V., Products of idempotent endomorphisms of relatively free algebras with weak exchange properties, Proc. Edinb. Math. Soc. 132(2007), 343–362.CrossRefGoogle Scholar
9. Fountain, J. and Lewin, A., Products of idempotent endomorphisms of an independence algebra of finite rank, Proc. Edinb. Math. Soc. 132(1992), 493–500.CrossRefGoogle Scholar
10. Fountain, J. and Lewin, A., Products of idempotent endomorphisms of an independence algebra of infinite rank, Math. Proc. Camb. Phil. Soc. 132(1993), 303–309.CrossRefGoogle Scholar
11. Fountain, J. B. and Petrich, M., Completely 0-simple semigroups of quotients, J. Alg. 101(1986), 365–402.CrossRefGoogle Scholar
12. Gould, V., Independence algebras, Alg. Univers. 33(1995), 294–318.CrossRefGoogle Scholar
13. Gould, V., Semigroups of left quotients: existence, uniqueness and locality, J. Alg. 267(2003), 514–541.CrossRefGoogle Scholar
14. Grätzer, G., Universal algebra(Van Nostrand, Princeton, NJ, 1968).Google Scholar
15. Howie, J. M., Fundamentals of semigroup theory(Oxford University Press, 1995).CrossRefGoogle Scholar
16. Laffey, T., Products of idempotent matrices, Linear Multilinear Alg. 14(1983), 309–314.CrossRefGoogle Scholar
17. McKenzie, R. N., McNulty, G. F. and Taylor, W. T., Algebra, lattices, varieties(Wadsworth, Florence, KY, 1983).Google Scholar
18. Marczewski, E., A general scheme of the notions of independence in mathematics, Bull. Acad. Polon. Sci. 6(1958), 731–736.Google Scholar
19. Narkiewicz, W., Independence in a certain class of abstract algebras, Fund. Math. 50(1961), 333–340.CrossRefGoogle Scholar
20. Ruitenberg, W., Products of idempotent matrices over Hermite domains, Semigroup Forum 46(1993), 371–378.CrossRefGoogle Scholar
21. Urbanik, K., Linear independence in abstract algebras, Colloq. Math. 132(1966), 233–255.CrossRefGoogle Scholar