Equations in the theory of Q-distributive lattices 外文电子书籍.pdf

DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 175 (1997) 211~ 19 Equations in the theory of Q-distributive lattices Alejandro Petrovich* Depurtamento de Matemgttica, Facultad de Cieneias Exactas y Naturales, Universidadde Buenos Aires, Ciudad Universitark~, 1428 Buenos Aires. Argentina Received 12 May 1995;revised 13 February 1996 Abstract 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 V y and V(x A V y ) =- Vx A Vy. A quantifier V is called simple if and only if it is given by the prescription: V0 -- 0 and V a = 1 for e a c h a ¢ 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-distr