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Equational bases for subvarieties of double MS-algebras

Published online by Cambridge University Press:  18 May 2009

T. S. Blyth ,
A. S. A. Noor  and
J. C. Varlet
T. S. Blyth
Affiliation:
Mathematical Institute, University of st Andrews, St Andrews KY16 9SS, Fife, Scotland.
A. S. A. Noor
Affiliation:
Institut de Mathematique, Universite de Liege, Avenue des Tilleuls, 15 B-400 Liege, Belgique.
J. C. Varlet
Affiliation:
Department of Mathematics, Rajshahi University, Rajshahi, Bangladesh.
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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 smallest element 0 and greatest element 1, and x ↦ x∘ is a unary operation such that l∘ = 0, x ≤ x∘∘ for all x ∈ L, and (x ∧ y)∘ = x∘ ∨ y∘ for all x, y ∈ L. These algebras belong to the class of Ockham algebras introduced by Berman [3]; see also [2,10,15]. A double MS-algebra is an algebra (L, ∨, ∧, ∘, +, 0, 1) of type (2, 2, 1, 1, 0, 0) such that (L, ∘) and (Ld, +) are MS-algebras, where Ld denotes the dual of L, and the operations ∘, + are linked by the identities x∘+ = x∘∘ and x+∘ = x++. We refer to [5, 6, 7, 8] for the basic properties of MS-algebras and double MS-algebras. Concerning the latter, the properties x∘∘∘ = x∘, x+++ = x+, and x∘ ≤ x+will be used frequently. The class of double MS-algebras is congruencedistributive and consequently the results of [13] can be applied. As to general results in lattice theory and universal algebra, the reader may consult [1, 9, 12].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 1 , January 1989 , pp. 1 - 16
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

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. Journal, 25 (1984), 175–181.Google Scholar
3.Berman, J., Distributive lattices with an additional unary operation, Aequationes Math., 16 (1977), 165–171.Google Scholar
4.Berman, Joel and Köhler, Peter, Finite distributive lattices and finite partially ordered sets, Mitt. Math. Sem. Giessen, 121 (1976), 103–124.Google Scholar
5.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
6.Blyth, T. S. and Varlet, J. C., Subvarieties of the class of MS-algebras, Proc. Roy. Soc. Edinburgh, 95A (1983), 157–169.Google Scholar
7.Blyth, T. S. and Varlet, J. C., Double MS-algebras, Proc. Roy. Soc. Edinburgh, 98A (1984), 37–47.CrossRefGoogle Scholar
8.Blyth, T. S. and Varlet, J. C., Subdirectly irreducible double MS-algebras, Proc. Roy. Soc. Edinburgh, 98A (1984), 241–247.Google Scholar
9.Burris, S. and Sankappanavar, H. P., A course in universal algebra, Graduate Texts in Mathematics 78 (Springer Verlag Berlin, 1981).Google Scholar
10.Cornish, W. H., Antimorphic Action, Research and Exposition in Mathematics, 12 (Heldermann Verlag Berlin, 1986).Google Scholar
11.Davey, B. A., On the lattice of subvarieties, Houston J. Math., 5 (1979), 183–192.Google Scholar
12.Grätzer, G., General Lattice Theory, (Birkhäuser Verlag Basel, 1978).Google Scholar
13.Jónsson, B., Algebras whose congruence lattices are distributive, Math. Scand., 21 (1967), 110–121.CrossRefGoogle Scholar
14.Noor, A. S. A., Bistable subvarieties of MS-algebras, Proc. Roy. Soc. Edinburgh, 105A (1987), 127–128.CrossRefGoogle Scholar
15.Sankappanavar, H. P., Distributive lattices with a dual endomorphism, Zeitschr. f. math. Logik und Grundlagen d. Math., 31 (1985), 385–392.CrossRefGoogle Scholar