Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T07:36:17.612Z Has data issue: false hasContentIssue false
  • English
  • Français

A simplistic approach to inverse transversals*

Published online by Cambridge University Press:  20 January 2009

T. S. Blyth  and
M. H. Almeida Santos
T. S. Blyth
Affiliation:
Mathematical Institute University of St Andrews, Scotland
M. H. Almeida Santos
Affiliation:
Departamento de Matemática F.C.T. Universidade Nova de Lisboa, Portugal
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.

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 1 , February 1996 , pp. 57 - 69
Copyright
Copyright © Edinburgh Mathematical Society 1996

Footnotes

*

NATO Collaborative Research Grant 910765 is gratefully acknowledged.

References

REFERENCES

1. Blyth, T. S. and McFadden, R., Naturally ordered regular semigroups with a greatest idempotent, Proc. Roy. Soc. Edinburgh 91A (1981), 107–122.CrossRefGoogle Scholar
2. Blyth, T. S. and McFadden, R., Regular semigroups with a multiplicative inverse transversal, Proc. Roy. Soc. Edinburgh 92A (1982), 253–270.CrossRefGoogle Scholar
3. Blyth, T. S. and Almeida, M. H. Santos, On naturally ordered regular semigroups with biggest idempotents, Comm. Algebra 21 (1993), 1761–1771.CrossRefGoogle Scholar
4. Blyth, T. S. and Santos, M. H. Almeida, Amenable orders on orthodox semigroups, J. Algebra 169 (1994), 49–70.CrossRefGoogle Scholar
5. Blyth, T. S. and Santos, M. H. Almeida, On weakly multiplicative inverse transversals, Proc. Edinburgh Math. Soc. 37 (1994), 91–99.CrossRefGoogle Scholar
6. Byleen, K., Regular four-spiral semigroups, idempotent semigroups and the Rees construction, Semigroup Forum 22 (1981), 97–100.CrossRefGoogle Scholar
7. McAlister, D. B. and McFadden, R., Regular semigroups with inverse transversals, Quart. J. Math. Oxford 34 (1983), 459–474.CrossRefGoogle Scholar
8. Pastijn, F., Rectangular bands of inverse semigroups, Simon Stevin 56 (1982), 1–97.Google Scholar
9. Saito, Tatsuhiko, Regular semigroups with a weakly multiplicative inverse transversal, in Proc. 8th Symposium on Semigroups (Shimane University, 1985), 22–25.Google Scholar
10. Saito, Tatsuhiko, Construction of regular semigroups with inverse transversals, Proc. Edinburgh Math. Soc. 32 (1989), 41–51.CrossRefGoogle Scholar
11. Tang, Xilin, Regular semigroups with inverse transversals, preprint.Google Scholar