Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T20:23:12.320Z Has data issue: false hasContentIssue false
  • English
  • Français

Injectives in some small varieties of ockham algebras

Published online by Cambridge University Press:  18 May 2009

R. Beazer
R. Beazer
Affiliation:
Department of Mathematics, University of Glasgow, Scotland
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 study of bounded distributive lattices endowed with an additional dual homomorphic operation began with a paper by J. Berman [3]. On the one hand, this class of algebras simultaneously abstracts de Morgan algebras and Stone algebras while, on the other hand, it has relevance to propositional logics lacking both the paradoxes of material implication and the law of double negation. Subsequently, these algebras were baptized distributive Ockham lattices and an order-topological duality theory for them was developed by A. Urquhart [13]. In an elegant paper [9], M. S. Goldberg extended this theory and, amongst other things, described the free algebras and the injective algebras in those subvarieties of the variety 0 of distributive Ockham algebras which are generated by a single finite subdirectly irreducible algebra. Recently, T. S. Blyth and J. C. Varlet [4] explicitly described the subdirectly irreducible algebras in a small subvariety MS of 0 while in [2] the order-topological results of Goldberg were applied to accomplish the same objective for a subvariety k1.1 of 0 which properly contains MS. The aim, here, is to describe explicitly the injective algebras in each of the subvarieties of the variety MS. The first step is to draw the Hasse diagram of the lattice AMS of subvarieties of MS. Next, the results of Goldberg are applied to describe the injectives in each of the join irreducible members of AMS. Finally, this information, in conjunction with universal algebraic results due to B. Davey and H. Werner [8], is applied to give an explicit description of the injectives in each of the join reducible members of AMS.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 2 , July 1984 , pp. 183 - 191
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

1.Balbes, R. and Dwinger, P., Distributive lattices, (University of Missouri Press, 1974).Google Scholar
2.Beazer, R., On some small varieties of distributive Ockham algebras, Glasgow Math. J. (to appear).Google Scholar
3.Berman, J., Distributive lattices with an additional unary operation, Aequationes Math. 16 (1977), 165–171.CrossRefGoogle Scholar
4.Blyth, T. S. and Varlet, J. C., On a common abstraction of de Morgan algebras and Stone algebras, Proc. Roy. Soc. Edinburgh Sect. A 94A (1983), 301–308.CrossRefGoogle Scholar
5.Blyth, T. S. and Varlet, J. C., Subvarieties of the class of MS-algebras, Proc. Roy. Soc. Edinburgh Sect. A 95A (1983), 157–169.CrossRefGoogle Scholar
6.Burris, S. and Sankappanavar, H. P., A course in universal algebra, (Springer-Verlag, 1981).CrossRefGoogle Scholar
7.Davey, B., On the lattice of subvarieties, Houston J. Math. 5 (1979), 183–192.Google Scholar
8.Davey, B. and Werner, H., Injectivity and Boolean powers, Math. Z. 166 (1979), 205–223.CrossRefGoogle Scholar
9.Goldberg, M. S., Distributive Ockham algebras: free algebras and injectivity, Bull. Austral. Math. Soc. 24 (1981), 161–203.CrossRefGoogle Scholar
10.Gratzer, G., General lattice theory, (Birkhauser Verlag, 1978).CrossRefGoogle Scholar
11.Priestley, H. A., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186–190.CrossRefGoogle Scholar
12.Priestley, H. A., Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. (3) 24, (1972), 507–530.CrossRefGoogle Scholar
13.Urquhart, A., Distributive lattices with a dual homomorphic operation, Studia Logica 38 (1979), 201–209.CrossRefGoogle Scholar