Free products of bounded distributive lattices

Abstract

The purpose of this note is to present an alternative approach to the main result of [4] by R. W. Quackenbush. Throughout, ~ will denote the equation class of bounded distributive lattices. The (bounded) extension A [B] of a universal algebra A by a Boolean algebra B (see [ 1; Ex. 63, p. 1 57]) may be defined to be the set ~ (32, A) of continuous maps from the Stone space X of B into the discrete space A, with the operations in ~(X, A) defined pointwise. It has been shown in [5] that when A is a bounded distributive lattice A [B] is isomorphic to A* B, the free product of A and B in ~3. In order to generalize this result, a definition of the extension D 1 [D2] of D~ by D 2 (Dr, D2s~)) is given in [4] as follows: Let 322 be the set of prime filters of D 2. Then Dt [D2] is the subalgebra of (D~) x~ generated by {fd ] dsD~} u {ga ] d~D2} where for all X~)['2, f~(x)=d and ga is the characteristic function of the set {xsX2 [ d~x}. Here it is shown that DI [D2] has a topological construction analogous to that for D a [B]. This provides an alternative proof for the fact that D~ [D2] -~Dt * D 2. We will utilize H. A. Priestley's duality theorem for bounded distributive lattices (see [2] and [3]): If D ~ ~3, then its set 32 of prime filters becomes a compact totally order disconnected space by identifying 32 with the set Horn z(D,2) of homomorphisms onto the two element chain; Hom z (D, 2) is a closed subspace of 2 ~ By the main result of [2], D is isomorphic to the lattice of all clopen increasing subsets of 32 under the isomorphism which sends d to {x~X] d~x}; alternatively, D is isomorphic to the lattice E~ (X,2) of continuous, monotone maps into the discrete space2 under the isomorphism which sends d to the characteristic function g,, of the set {xeX ] dsx} .