Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T00:41:39.208Z Has data issue: false hasContentIssue false
  • English
  • Français

The Translational Hull of an Inverse Semigroup

Published online by Cambridge University Press:  20 November 2018

N. R. Reilly
N. R. Reilly*
Affiliation:
Simon Fraser University, Burnaby, British Columbia
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.

Let S be a semigroup. A function λ(ρ) on S is a left (right) translation of S if, for all x, y ∊ S, λ(xy) = λ(x)y ((xy)ρ= x(yρ)). A left translation λ and a right translation ρ are said to be linked if x(λy) = (xρ)y, for all x,y ∊ S,and then the ordered pair (λ, ρ) is called a bitranslation. Clearly the set Λ(S) (P(S)) of all left (right) translations is a semigroup with respect to composition of functions. The set of bitranslations forms a subsemigroup of the direct product Λ(S) × P(S)which is called the translational hull, Ω(S), of S. A valuable survey of results relating to Ω(S) and its importance in relation to semigroup extensions will be found in Petrich's review [6], to which the reader is referred for basic results on translational hulls.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 5 , October 1974 , pp. 1050 - 1068
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Ault, J. E., Translational hull of an inverse semigroup, Semigroup Forum 4 (1972), 165–168.Google Scholar
2. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. 1, Math. Surveys No. 7 (Amer. Math. Soc, Providence, R. I., 1962).Google Scholar
3. Gluskin, L. M., Ideals of semigroups, Mat. Sb. 55 (1961), 421–448.Google Scholar
4. Howie, J. M., The maximum idempotent separating congruence on an inverse semigroup, Proc. Edinburgh Math. Soc. 14 (1964), 71–79.Google Scholar
5. Munn, W. D., Fundamental inverse semigroups, Quart. J. Math. Oxford Ser. 21 (1970), 157-170.Google Scholar
6. Petrich, M., The translational hull in semigroups and rings, Semigroup Forum 1 (1970), 283–360.Google Scholar
7. Munn, W. D., The translational hull of a completely 0-simple semigroup, Glasgow Math. J. 9 (1968), 1–11.Google Scholar
8. Ponizovski, I. S., A remark on inverse semigroups, Uspehi Math. Nauk. 20 (1965), 147–148.Google Scholar
9. Reilly, N. R. and Scheiblich, H. E., Congruences on regular semigroups, Pacific J. Math. 23 (1967), 349–360.Google Scholar