Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T11:37:05.147Z Has data issue: false hasContentIssue false
  • English
  • Français

On regular semigroups whose idempotents form a subsemigroup

Published online by Cambridge University Press:  17 April 2009

T. E. Hall
T. E. Hall
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria.
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.

For brevity the semigroups in the title are called orthodox semigroups. The finest inverse semigroup congruence on an orthodox semigroup is shown to have a simple form and conversely, regular semigroups whose finest inverse congruence has this simple form are shown to be orthodox. Next ideal extensions of orthodox semigroups by orthodox semigroups are shown to be also orthodox, whence a finite semigroup is orthodox if and only if each principal factor is orthodox and completely O-simple or simple. Finally it is determined which completely O-simple semigroups are orthodox.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 2 , August 1969 , pp. 195 - 208
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Clifford, A.H. and Preston, G.B., The algebraic theory of semigroups, vols. 1 and 2, (Amer. Math. Soc. Mathematical Surveys 7, Providence, 1961 and 1967).Google Scholar
[2]Fantham, P.H.H., “On the classification of a certain type of semigroup”, Proc. London Math. Soc. (3) 10 (1960), 409–427.CrossRefGoogle Scholar
[3]Howie, J.M. and Lallement, G., “Certain fundamental congruences on a regular semigroup”, Proc. Glasgow Math. Assoc. 7 (1965–1966), 145–149.CrossRefGoogle Scholar
[4]Kimura, Naoki and Yamada, Miyuki, “Note on idempotent semigroups II”, Proc. Japan Acad., 34 (1958), 110–112.Google Scholar
[5]Lallement, G., “Congruences et équivalences de Green sur un demi-groupe régulier”, C.R. Acad. Sci. Paris, Série A, 262 (1966), 613–616.Google Scholar
[6]Meakin, J.C., “Congruences on orthodox semigroups”, J. Austral. Math. Soc. (to appear).Google Scholar
[7]Reilly, N.R. and Scheiblich, H.E., “Congruences on regular semigroups”, Pacific J. Math. 23 (1967), 349–360.CrossRefGoogle Scholar
[8]Tamura, Takayuki, “Decompositions of a completely simple semigroup”, Osaka Math. J. 12 (1960), 269–275.Google Scholar
[9]Yamada, Miyuki, “Regular semigroups whose idempotents satisfy permutation identities”, Pacific J. Math. 21 (1967), 371–392.CrossRefGoogle Scholar