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The ideal lattice of an MS-algebra†

Published online by Cambridge University Press:  18 May 2009

T. S. Blyth  and
J. C. Varlet
T. S. Blyth
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Fife, Scotland
J. C. Varlet
Affiliation:
Institut de Mathématique, Université de Liège, Avenue des tilleuls, 15, B-4000 Liège, Belgique
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Recently we introduced the notion of an MS-algebra as a common abstraction of a de Morgan algebra and a Stone algebra [2]. Precisely, an MS-algebra is an algebra 〈L; ∧, ∨ ∘, 0, 1〉 of type 〈2, 2, 1, 0, 0〉 such that 〈L; ∧, ∨, 0, 1〉 is a distributive lattice with least element 0 and greatest element 1, and x → x∘ is a unary operation such that x ≤ x∘∘, (x ∧ y)∘ = x∘ ∨ y∘ and 1∘ = 0. It follows that ∘ is a dual endomorphism of L and that L∘∘ = {x∘∘ x ∊ L} is a subalgebra of L that is called the skeleton of L and that belongs to M, the class of de Morgan algebras. Clearly, theclass MS of MS-algebras is equational. All the subvarieties of MS were described in [3]. The lattice Λ (MS) of subvarieties of MS has 20 elements (see Fig. 1) and its non-trivial part (we exclude T, the class of one-element algebras) splits into the prime filter generated by M, that is [M, M1], the prime ideal generated by S, that is [B, S], and the interval [K, K2 ∨ K3].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 2 , May 1988 , pp. 137 - 143
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Birkhoff, G., Lattice theory, Amer. Math. Soc. Coll. Publ. 25 (1967).Google Scholar
2.Blyth, T. S. and Varlet, J. C., On a common abstraction of de Morgan algebras and Stone algebras, Proc. Roy. Soc. Edinburgh 94A (1983), 301–308.CrossRefGoogle Scholar
3.Blyth, T. S. and Varlet, J. C., Subvarieties of the class of MS-algebras, Proc. Roy. Soc. Edinburgh 95A (1983), 157–169.Google Scholar
4.Blyth, T. S. and Varlet, J. C., Fixed points in MS-algebras, Bull. Soc. Roy. Sci. Liège 53 (1984), 3–8.Google Scholar