Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T01:41:00.271Z Has data issue: false hasContentIssue false
  • English
  • Français

The lattice of full regular subsemigroups of a regular semigroup

Published online by Cambridge University Press:  14 November 2011

Katherine G. Johnston  and
Peter R. Jones
Katherine G. Johnston
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, U.S.A.
Peter R. Jones
Affiliation:
Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, Wisconsin 53233, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Although the regular subsemigroups of a regular semigroup S do not, in general, form a lattice in any naturalway, it is shown that the full regular subsemigroups form a complete sublattice LF of the lattice of all subsemigroups; moreover this lattice has many of the nice features exhibited in (the special case of) the lattice of full inverse subsemigroups of an inverse semigroup, previously studied by one of the authors. In particular, LF is again a subdirect product of the corresponding lattices for each of the principal factors of S.

A description of LF for completely 0-simple semigroups is given. From this, lattice-theoretic properties of LF may be found for completely semisimple semigroups. For instance, for any such combinatorial semigroup, LF is semimodular.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 98 , Issue 3-4 , 1984 , pp. 203 - 214
Copyright
Copyright © Royal Society of Edinburgh 1984

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. II (Math. Surveys 7) (Providence, R.I.: Amer. Math. Soc, 1967).Google Scholar
2Crawley, P. and Dilworth, R. P.. Algebraic theory of lattices (Englewood Cliffs, N.J.: Prentice-Hall, 1973).Google Scholar
3Eberhart, C., Williams, W. and Kinch, L.. Idempotent-generated regular semigroups. J. Austral. Math. Soc. 15 (1973), 27–34.CrossRefGoogle Scholar
4Fitzgerald, D. G.. On inverses of products of idempotents in regular semigroups. J. Austral. Math. Soc. 13 (1972), 335–337.CrossRefGoogle Scholar
5Hall, T. E.. On regular semigroups. J. Algebra 24 (1973), 1–24.CrossRefGoogle Scholar
6Howie, J. M.. An introduction to semigroup theory (London: Academic Press, 1976).Google Scholar
7Howie, J. M.. Idempotents in completely 0-simple semigroups. Glasgow Math. J. 19 (1978), 109–113.CrossRefGoogle Scholar
8Johnston, K. G.. Subalgebra lattices of completely simple semigroups. Semigroup Forum 29 (1984), 109–121.CrossRefGoogle Scholar
9Jones, P. R.. Semimodular inverse semigroups. J. London Math. Soc. 17 (1978), 446–456.CrossRefGoogle Scholar
10Jones, P. R.. Distributive inverse semigroups. J. London Math. Soc. 17 (1978), 457–466.CrossRefGoogle Scholar
11Nambooripad, K. S. S.. The natural partial order on a regular semigroup. Proc. Edinburgh Math. Soc. 23 (1980), 249–260.CrossRefGoogle Scholar
12Pastijn, F.. Idempotent-generated completely 0-simple semigroups. Semigroup Forum 15 (1977), 41–50.CrossRefGoogle Scholar
13Sevrin, L. N. and Ovsyannikov, A. J.. Semigroups and their subsemigroup lattices. Semigroup Forum 27 (1983), 1–154.CrossRefGoogle Scholar
14Suzuki, M.. Structure of a group and the structure of its lattice of subgroups (Berlin: Springer, 1956).CrossRefGoogle Scholar