Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T01:01:12.051Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Conjecture Concerning Semigroup Homomorphisms

Published online by Cambridge University Press:  20 November 2018

R. J. Plemmons
R. J. Plemmons*
Affiliation:
The University of Tennessee, Knoxville, Tennessee
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.

In this paper we settle (with a counterexample) the question raised by Clifford and Preston in [2, p. 275], concerning maximal group homomorphic images of semigroups. We also consider the question in a more general context and characterize all such examples. The notation and definitions follow [1; 2].

By a type of semigroups we mean a class of semigroups, closed under isomorphisms and containing the one-element semigroup. If S is any semigroup and is a type, then a semigroup S* is defined, in [1, p. 18], to be a maximal homomorphic image of S having type if

(i) ,

(ii) S* is a homomorphic image of S, and

(iii) whenever and T is a homomorphic image of S, then there exists a homomorphism from S* onto T.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 641 - 644
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Clifford, A. H. and Preston, G. P., The algebraic theory of semigroups, Vol. I, Mathematical Surveys, No. 7 (Amer. Math. Soc, Providence, R. I., 1961).Google Scholar
2. Clifford, A. H. and Preston, G. P., The algebraic theory of semigroups, Vol. II, Mathematical Surveys, No. 7 (Amer. Math. Soc, Providence, R. I., 1967).Google Scholar
3. Good, R. A. and Hughes, D. R., Associated groups for a semigroup, Bull. Amer. Math. Soc. 58 (1952), 264–265.Google Scholar
4. McAlister, D. B., A homomorphism theorem for semigroups, J. London Math. Soc. 43 (1968), 355–366.Google Scholar
5. Plemmons, R. J. and Tamura, T., Semigroups with a maximal homomorphic image having zero, Proc. Japan Acad. 41 (1965), 681–685.Google Scholar
6. Stoll, R. R., Homomorphisms of a semigroup onto a group, Amer. J. Math. 73 (1951), 475–481.Google Scholar
7. Tamura, T., Maximal or greatest homomorphic images of a given type, Can. J. Math. 20 (1968), 264–271.Google Scholar