Hostname: page-component-76c49bb84f-ndtt8 Total loading time: 0 Render date: 2025-07-04T12:13:44.642Z Has data issue: false hasContentIssue false
  • English
  • Français

Divisibility in categories of a class which includes the category of binary relations

Published online by Cambridge University Press:  18 May 2009

D. G. Fitz-Gerald
D. G. Fitz-Gerald
Affiliation:
University of QueenslandSt. Lucia, Queensland
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.

Green's relations are essential for “co-ordinatizing” semigroups. Jacqueline Klasa, in applying cognate ideas to categories [4, 5], has shown that divisibility in suitably-behaved categories may be described in terms of subobjects and quotients.

Here it is shown that adjoint functors which are onto objects preserve divisibility (in a certain sense). The inclusion functor of the category of sets into the category R of binary relations is such a functor. A slight modification of its right adjoint allows the representation of R as a full subcategory in a category CSL of complete semilattice morphisms.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 17 , Issue 1 , January 1976 , pp. 22 - 30
Copyright
Copyright © Glasgow Mathematical Journal Trust 1976

References

REFERENCES

1.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Volume 1, Amer. Math. Soc. Math. Surveys No. 7 (Providence, 1961).CrossRefGoogle Scholar
2.Fitz-Gerald, D. G. and Preston, G. B., Divisibility of binary relations, Bull. Austral. Math. Soc. 5 (1971), 75–86.CrossRefGoogle Scholar
3.Isbell, J. R., Some remarks concerning categories and subspaces, Can. J. Math. 9 (1957), 563–577.CrossRefGoogle Scholar
4.Klasa, J., Équivalences de Green dans les catégories bien filtrantes, Séminaire Dubreil — Pisot, 20e année (1967), Fasc. 2, no. 17.Google Scholar
5.Klasa, J., Semisimplicity and von Neumann's regularity, Semigroup Forum 2 (1971), 354–361.CrossRefGoogle Scholar
6.MacLane, S., Categories for the working mathematician, Graduate Texts in Mathematics 5 (New York, 1971).Google Scholar
7.Preston, G. B., Embedding any semigroup in a -simple semigroup, Trans. Amer. Math. Soc. 93 (1959), 351–355.Google Scholar
8.Reilly, N. R., Embedding inverse semigroups in bisimple inverse semigroups, Quart. J. Math. Oxford (2) 16 (1965), 183–187.CrossRefGoogle Scholar
9.Yang, Jaw-Ching, A theorem on the semigroup of binary relations, Proc. Amer. Math. Soc. 22 (1969), 134–135.CrossRefGoogle Scholar
10.Zaretskii, K. A., Polugruppa binarnikh otnoshenii [The semigroup of binary relations], Mat. Sb. (N.S.) 61 (103) (1963), 291–305.Google Scholar