The countable chain condition and free algebras

Abstract

Let A be an algebra that contains a constant 0 and a binary operation 9 among its fundamental operations. Elements x, yeA are called disjoint if x . y = 0 . A satisfies the countable chain condition (c.c.c.) if it contains no uncountable set of pairwise disjoint nonzero elements. Consider the following facts: 1. (Classical) A free Boolean algebra satisfies c.c.c. 2. (Abian [-1]) A free algebra in the ring variety generated by a finite field satisfies c.c.c. These examples suggest the question ~ c.c.c, hold in every free algebra of a quasi-primal variety?" Before this question can be answered we must know what the c.c.c, means in a universal algebra. In Section 2 we propose a general algebraic condition, called the countable separability condition (c.s.c.), and show that, in many situations, it is equivalent to c.c.c. Using c.s.c, we give an affirmative answer to the posed question (Theorem 2.9). In Section 1 the free algebras in quasi-primal varieties are determined. The main tool for their description is the second duality result for quasi-primal varieties due to Keimel and Werner [2]. For unexplained notation and terminology see [2].