Hostname: page-component-6d856f89d9-fb4gq Total loading time: 0 Render date: 2024-07-16T08:47:42.934Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of One Primitive Inverse Semigroup by Another

Published online by Cambridge University Press:  20 November 2018

Janet E. Ault
Janet E. Ault*
Affiliation:
University of Florida, Gainesville, Florida
Rights & Permissions [Opens in a new window]

Extract

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.

Every inverse semigroup containing a primitive idempotent is an ideal extension of a primitive inverse semigroup by another inverse semigroup. Consequently, in developing the theory of inverse semigroups, it is natural to study ideal extensions of primitive inverse semigroups (cf. [3; 7]). Since the structure of any primitive inverse semigroup is known, an obvious type of ideal extension to consider is that of one primitive inverse semigroup by another. In this paper, we will construct all such extensions and give an abstract characterization of the resulting semigroup.

The problem of extending one primitive inverse semigroup by another can be essentially reduced to that of extending one Brandt semigroup by another Brandt semigroup. The latter problem has been solved by Lallement and Petrich in [3] in case the first Brandt semigroup has only a finite number of idempotents.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 270 - 278
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. 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
2. Gérard, Lallement and Mario, Petrich, Décompositions I-matricielles d'un demi-groupe, J. Math. Pures Appl. 45 (1966), 67–117.Google Scholar
3. Gérard, Lallement and Mario, Petrich, Extensions of a Brandt semigroup by another, Can. J. Math. 22 (1970), 974–983.Google Scholar
4. Mario, Petrich, The translational hull of a completely 0-simple semigroup, Glasgow Math. J. 9 (1968), 1–11.Google Scholar
5. Mario, Petrich, Translational hull and semigroups of binary relations, Glasgow Math. J. 9 (1968), 12–21.Google Scholar
6. Gábor, Szász, Introduction to lattice theory (Academic Press, New York, 1963).Google Scholar
7. Warne, R. J., Extensions of Brandt semigroups and applications, Illinois J. Math. 10 (1966), 652–660.Google Scholar