Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-21T22:41:36.483Z Has data issue: false hasContentIssue false
  • English
  • Français

Notes on congruences on regular semigroups.I

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

R. J. Koch  and
B. L. Madison
R. J. Koch
Affiliation:
Louisiana State University Baton Rouge, Louisiana 70803, U.S.A.
B. L. Madison
Affiliation:
University of Arkansas Fayetteville, Arkansas 72701, U.S.A.
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.

Four properties of congruences on a regular semigroup S are studied and compared. Let R, L and D denote Green's relations and let V = {(a, b) ∈ S × S|a and b are mutually inverse}. A congruence ρ on S is (1) rectangular provided ρ ∩ D = (ρ ∩ L) ° (ρ ∩ R), (2) V-commuting provided ρ ° V = V ° ρ, (3) (L, R)-commuting provided L ° ρ = ρ ° L, and R ° ρ = ρ ° R, and (4) idempotent-regular provided each idempotent ρ-class is a regular subsemigroup of S.

A rectangular congruence is (L, R)-commuting and a V-commuting congruence is idempotent-regular. If ρ is idempotent-regular and (L, R)-commuting then ρ is V-commuting. Examples and conditions are given to show what other implications among the four properties hold. In addition to characterizations of the properties, these are studied in the presence of other conditions on S. For example, if S is a stable regular semigroup, then each congruence under D is rectangular.

MSC classification

Secondary: 20M15: Mappings of semigroups 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 39 , Issue 2 , October 1985 , pp. 146 - 158
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Anderson, L. W., Hunter, R. P. and Koch, R. J., ‘Some results on stability in semigroups’, Trans. Amer. Math. Soc. 117 (1965), 521–529.CrossRefGoogle Scholar
[2]Byleen, K., Meakin, J. and Pastijn, F., ‘The fundamental four-spiral semigroup’, J. Algebra 54 (1978), 6–26.Google Scholar
[3]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Math. Surveys No. 7 (Amer. Math. Soc., Providence, R. I., Vol. I, 1961, Vol. II, 1967).Google Scholar
[4]Hall, T. E., ‘Congruences and Green's relations on regular semigroups’, Glasgow Math. J. 13 (1972), 167–175.Google Scholar
[5]Hall, T. E., ‘On regular semigroups’, J. Algebra 24 (1973), 1–24.CrossRefGoogle Scholar
[6]Koch, R. J. and Madison, B. L., ‘Congruences under L on regular semigroups’, Simon Stevin 57 (1983), 273–283.Google Scholar
[7]Koch, R. J., ‘Comparison of congruences on regular semigroups’, Semigroup Forum 26 (1983), 295–305.CrossRefGoogle Scholar
[8]Koch, R. J., ‘On inverse of products in regular semigroups’, preprint.Google Scholar
[9]Nambooripad, K. S. S., ‘Structure on regular semigroups’, Symposium on regular semigroups, pp. 101–117 (Northern Illinois University, DeKalb, Illinois, 1979).Google Scholar
[10]Nambooripad, K. S. S. and Pastijn, F., ‘V-regular semigroups’, preprint.Google Scholar
[11]Onstad, J. A., A study of certain classes of regular semigroups (Dissertation, University of Nebraska-Lincoln, 1974).Google Scholar
[12]Reilly, N. R. and Scheiblich, H. E., ‘Congruences on regular semigroups’, Pacific J. Math. 23 (1967), 349–360.Google Scholar