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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

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