Discriminator varieties of Boolean algebras with residuated operators

Abstract

The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral ralgebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated monoids are discriminator algebras, provided they are subdirectly irreducible. These results are then used to give equational bases for some varieties of r-algebras. We also show that the variety of all residuated Boolean monoids is not a discriminator variety, which answers a question of B. Jonsson. 1. Preliminaries. A unary operation f on a Boolean algebra A0 = (A,+, 0, ·, 1,− ) is additive if f(x + y) = f(x) + f(y) and normal if f(0) = 0. For an n-ary operation f on A0, a sequence a ∈ A and i < n we define the (a, i)-translate of f to be the unary operation fa,i(x) = f(a0, . . . , ai−1, x, ai+1, . . . , an−1) . An operator on A0 is an n-ary operation for which all (a, i)-translates are additive and normal. Note that 0-ary operations (constants) have no translates, so they are operators by default. A = (A0,F) is a Boolean algebra with operators (BAO for short) if each f ∈ F is an operator on A0. The arity (or rank) of f is denoted by %f . To be an operator on a Boolean algebra is of course an equational property, and the variety of all BAOs with operators in F will be denoted by BAOF . The variety BAO{f}, where f is a unary operator, is usually referred to as the variety of modal algebras (the algebraic counterpart of modal logic). 1991 Mathematics Subject Classification: Primary 06E25; Secondary 03G15, 08A40, 03B45. The paper is in final form and no version of it will be published elsewhere.