THE STRUCTURE OF PSEUDOCOMPLEMENTED DISTRIBUTIVE LATTICES. II: CONGRUENCE EXTENSION AND AMALGAMATION
Amal Amal · 2016
This paper continues the examination of the structure of pseudocomplemented distributive lattices. First, the Congruence Extension Property is proved. This is then applied to examine properties of the equational classes A,-1, n<, which is a complete list of all the equational classes of pseudocomplemented distributive lattices (see Part I). The (i.e., the semigroup generated by the operators H, S, and P) are described. The Amalgamation Property is shown to hold iff n< 2 or n = w. For 3< n < co, does not satisfy the Amalgamation Property; the deviation is measured by a class Amal (,kn) (C 2n The finite algebras in Amal (2n) are determined. 0. Introduction. This paper continues the examination of the structure of pseudocomplemented distributive lattices begun in Part I, [8]. Using the description of congruences given in Part I, we verify the Congruence Extension Property in ?1. This, in effect, states that a *-congruence on a subalgebra can be extended to a *-congruence on the algebra. This property is applied in ??2 and 3. In ?2 we determine the standard semigroups of the equational classes of pseudocomplemented distributive lattices, which is, roughly speaking, the semigroup generated by the operators H, S, and P in the sense of [5]. In ?3 it is shown that the Amalgamation Property holds in 4iOn (notation of Part I) if and only if n = -1, 0, 1, 2, or co, and that the subalgebra theorem for free products of B. Jonsson [7] holds for exactly the same equational classes. Since the Amalgamation Property fails to hold for 3, 4, . . ., we introduce a concept attempting to measure the extent of this failure. This concept is the amalgamation class of M Amal (Xk). The Amalgamation Property holds in -X if and only if Amal ( =) = ?4 contains results on Amal(n) for 2< n < c; in particular, we determine the finite algebras in Amal (?J.) 1. The Congruence Extension Property. A class Xk of algebras is said to satisfy the Congruence Extension Property if, given any algebra B and subalgebra A, both in V and any congruence E) on A, there is a congruence 0 on B such that the Received by the editors July 8, 1970. AMS 1970 subject classifications. Primary 06A35; Secondary 08A25, 18A20, 18A30, 18C05.