Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T21:26:08.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Injectivity and Congruence Extension in Congruence-Distributive Equational Classes

Published online by Cambridge University Press:  20 November 2018

Brian A. Davey
Brian A. Davey*
Affiliation:
La Trobe University, Bundoora, Victoria, Australia
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.

There are many concepts which arise naturally in a discussion of injectivity in an equational class; for example, weak injective algebras, absolute subretracts, essential extensions, the congruence extension property, and the amalgamation property (see [3; 9; 17; 18]). It has already been demonstrated in several papers, notably [9; 17; 26; 27; 28], that the study of these concepts is greatly enriched by the assumption that the algebras under consideration have distributive congruence lattices. In this work attention is focused on weak injective algebras (Section 2) and the congruence extension property (Section 3).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 3 , 01 June 1977 , pp. 449 - 459
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Balbes, R., Projective and injective distributive lattices. Pacific J. Math. 21 (1967), 405–435.Google Scholar
2. Balbes, R. and G. Grâtzer, Injective and projective Stone algebras, Duke Math. J. 38 (1971), 339–347.Google Scholar
3. Banaschewski, B., Injectivity and essential extensions in universal algebra, Proc. Conf. in Universal Algebra, Queen's Univ., Kingston, Ont., 1969 (Queen's Univ., Kingston, Ont., 1970), pp. 131–147.Google Scholar
4. Bruns, G. and Kalmbach, G., Varieties of orthomodular lattices, Can. J. Math. 23 (1971), 802–810.Google Scholar
5. Bruns, G. and Kalmbach, G. Varieties of orthomodular lattices, II, Can. J. Math. 24 (1972), 328–337.Google Scholar
6. Cignoli, R., Infective De Morgan and Kleene algebras, Proc. Amer. Math. Soc. 47 (1975), 269–278.Google Scholar
7. Davey, B. A., Weak infectives and Boolean extensions, Notices Amer. Math. Soc, 22 (1975), A-618. Abstract #75T-A224.Google Scholar
8. Davey, B. A. Dualities for equational classes of Brouwerian algebras and Heyting algebras, Trans. Amer. Math. Soc. 221 (1976), 119–146.Google Scholar
9. Day, A., Injectivity in equational classes of algebras, Can. J. Math. 24 (1972), 209–220.Google Scholar
10. Day, A. The congruence extension property and subdirectly irreducible algebras—an example, Algebra Univ. 3 (1973), 229–237.Google Scholar
11. Frayne, T., Morel, A. C., and Scott, D., Reduced direct products, Fund. Math. 51 (1962), 195–228.Google Scholar
12. Fried, E., Tournaments and nonassociative lattices, Ann. Univ. Sci. Budapest.Eôtvôs Sect. Math. 13 (1970), 151–164.Google Scholar
13. Fried, E. Weakly associative lattices with congruence extension property, Algebra Univ. 4 (1974), 151–162.Google Scholar
14. Fried, E. and Gràtzer, G., A nonassociative extension of the class of distributive lattices, Pacific J. Math. 49 (1973), 59–78.Google Scholar
15. Fried, E., Gràtzer, G., and Lakser, H., Amalgamation and weak infectives in the equational class of modular lattices Mn, Notices Amer. Math. Soc. 18 (1971), 624. Abstract #71TA101.Google Scholar
16. Gràtzer, G., Universal algebra (Van Nostrand, Princeton, N. J., 1968).Google Scholar
17. Gràtzer, G. and Lakser, H., The structure of pseudocomplemented distributive lattices. II. Congruence extension and amalgamation, Trans. Amer. Math. Soc. 156 (1971), 343–358.Google Scholar
18. Gràtzer, G. and Lakser, H. The structure of pseudocomplemented distributive lattices. III. Infectives and absolute subretracts, Trans. Amer. Math. Soc. 169 (1972), 475–487.Google Scholar
19. Halmos, P. R., Algebraic logic (Chelsea, New York, 1962).Google Scholar
20. Halmos, P. R. Lectures on Boolean algebras, Math. Studies No. 1. (Van Nostrand, Princeton, N. J., 1963).Google Scholar
21. Henkin, L. and Tarski, A., Cylindric algebras, in Lattice Theory, Proc. of Symp. in Pure Math., II (Amer. Math. Soc, Providence, R. I., 1961), pp. 83–113.Google Scholar
22. Jônsson, B., Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110–121.Google Scholar
23. T., Katrinâk, Infective double Stone algebras, Algebra Univ. 4 (1974), 259–267.Google Scholar
24. Congruence extension property for distributive double p-algebras, Algebra Univ. 4 (1974), 273–276.Google Scholar
25. Lakser, H., The structure of pseudocomplemented distributive lattices. I. Subdirect decomposition, Trans. Amer. Math. Soc. 156 (1971), 334–342.Google Scholar
26. Quackenbush, R. W., Demi-semi-primal algebras and MaVcev type conditions, Math. Z. 122 (1971), 166–176.Google Scholar
27. Quackenbush, R. W. Structure theory for equational classes generated by quasi-primal algebras, Trans. Amer. Math. Soc. 187 (1974), 127–145.Google Scholar
28. Quackenbush, R. W. Semi-simple equational classes with distributive congruence lattices, Ann. Univ. Sci. Budapest.Eôtvôs Sect. Math. 17 (1974), 15–19.Google Scholar
29. Rasiowa, H. and Sikorski, R., The mathematics of metamathematics (Monag. Mat. Vol. 41, Panstowe Wydawn, Nankowe, Warsaw, 1963).Google Scholar