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Equations in the theory of Q-distributive lattices

✍ Scribed by Alejandro Petrovich

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
459 KB
Volume
175
Category
Article
ISSN
0012-365X

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✦ Synopsis

A Q-distributive lattice is an algebra (L, v, A, V, 0, 1 ) of type (2, 2, 1, 0, 0) such that (L, V, A, 0, 1 ) is a bounded distributive lattice and 27 satisfies the equations V0 = 0, x A Vx = x, V(x V y) = Vx V Vy and V(x A Vy) = Vx A 27y. The aim of this paper is to find, for each proper subvariety of the variety of Q-distributive lattices, an equation which determines it, relatively to the whole variety, as well as to give a characterization of the minimum number of variables needed in such equation.

A quantifier on a bounded distributive lattice L is a unary operation V on L that satisfies V0 = 0, x A Vx = x, V(x V y) = Vx V Vy and V(x A Vy) =-Vx A Vy. A quantifier V is called simple if and only if it is given by the prescription: V0 --0 and Va= 1 for eachaΒ’0.

A Q-distributive lattice is an algebra (L, V, A, V, 0, 1) of type (2, 2, 1, 0, 0) such that (L, V, A, 0, 1 ) is a bounded distributive lattice and V is a quantifier on L. The variety of Q-distributive lattices will be denoted by 2.

Q-distributive lattices were introduced by Cignoli in [2] and he showed that the lattice of equational subclasses of ~ is a chain of type (o + 1.

In this paper we first find, for each proper subvariety of ~, an equation which determines it. Secondly, we give a characterization of the minimum number of variables needed in an equation characterizing a given subvariety, and we determine this number in some cases.

We begin with some notation. We shall denote by 2 the Boolean algebra with two elements. For each natural number p, Bp will denote the Boolean algebra 2 p endowed with the simple quantifier, and Cp the lattice 2 p with a new 1 added endowed with the simple quantifier, provided p~> 1, and Co = B0. For each (p, q) E w x w, Opq will denote the subjacent lattice Bp x Cq endowed with the simple quantifier. (Observe, in

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