Hostname: page-component-6d856f89d9-nr6nt Total loading time: 0 Render date: 2024-07-16T06:38:23.570Z Has data issue: false hasContentIssue false
  • English
  • Français

Injectivity in Equational Classes of Algebras

Published online by Cambridge University Press:  20 November 2018

Alan Day
Alan Day*
Affiliation:
Vanderbilt University, Nashville, Tennessee; Lakehead University, Thunder Bay, Ontario
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.

The concept of injectivity in classes of algebras can be traced back to Baer's initial results for Abelian groups and modules in [1]. The first results in non-module types of algebras appeared when Halmos [14] described the injective Boolean algebras using Sikorski's lemma on extensions of Boolean homomorphisms [19]. In recent years, there have been several results (see references) describing the injective algebras in other particular equational classes of algebras.

In [10], Eckmann and Schopf introduced the fundamental notion of essential extension and gave the basic relations that this concept had with injectivity in the equational class of all modules over a given ring. They developed the notion of an injective hull (or envelope) which provided every module with a minimal injective extension or equivalently, a maximal essential extension. In [6] and [9], it was noted that these relationships hold in any equational class with enough injectives.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 2 , April 1972 , pp. 209 - 220
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Baer, R., Abelian groups that are direct summands of every containing Abelian group, Bull. Amer. Math. Soc. 46 (1940), 800–806.Google Scholar
2. Balbes, R., Projective and injective distributive lattices, Ph.D. dissertation, University College of Los Angeles, 1966.Google Scholar
3. Balbes, R. and Gratzer, G., Injective and projective Stone algebras, Prepublication manuscript.Google Scholar
4. Balbes, R. and Horn, A., Injective and projective Heyting algebras, Trans. Amer. Math. Soc. 148 (1970), 549–560.Google Scholar
5. Banaschewski, B., Projective covers in categories of topological spaces and topological algebras, Proceedings of the Topological Conference, Kanpur (1968).Google Scholar
6. Banaschewski, B., Injectivity and essential extensions in equational classes, Proceedings of the Conference in Universal Algebra, Queens University (1969).Google Scholar
7. Banaschewski, B. and Bruns, G., Categorical characterization of the MacNeille completion, Arch. Math. (Basle) 18 (1967), 369–377.Google Scholar
8. Banaschewski, B. and Bruns, G., Injective hulls in the category of distributive lattices, J. Reine Angew Math. 232 (1967), 102–109.Google Scholar
9. Day, A., Injectives in non-distributive equational classes of lattices are trivial, Arch. Math. (Basle) 21 (1970), 113–115.Google Scholar
10. Eckmann, B. und Schopf, A., Uber injective Modulen, Arch. Math. (Basle) 4 (1953), 75–78 Google Scholar
11. Gould, M. and Gratzer, G., Boolean extensions and normal subdirect powers in finite universal algebras, Math. Z. 99 (1967), 16–25.Google Scholar
12. Gratzer, G., Universal algebra (Van Nostrand, Princeton, N.J., 1968).Google Scholar
13. Gratzer, G. and Lakser, H., The structure of pseudo-complemented distributive lattices. II, Prepublication manuscript.Google Scholar
14. Halmos, P. R., Injective and projective Boolean algebras, Proc. Sympos. Pure Math. 11 (1961), 114–122.Google Scholar
15. Jónsson, B., Algebras whose congruence lattice is distributive, Math. Scand. 21 (1967), 110–121.Google Scholar
16. Lee, K. B., Equational Classes of distributive pseudo-complemented lattices (to appear in Can. J. Math.).Google Scholar
17. Michler, G. and Wille, R., Die primitiven Klassen arithmetischen Ringe, Math. Z. 113 (1970), 369–372.Google Scholar
18. Mitchell, B., Theory of categories (Academic Press, New York, 1965).Google Scholar
19. Sikorski, R., A theorem on extension of homomorphisms, Ann. Polon. Math. 21 (1948), 332–335.Google Scholar