A Modification of Polin’s Variety

In this note we settle a question posed by Hobby and McKenzie in [2] on the nature of locally finite equational classes which satisfy some nontrivial congruence identity. We settle problem #14 from [2] by exhibiting a locally finite equational class V which omits types 1 and 5 and which satisfies some non-trivial congruence identity, but which contains a finite algebra having a type 4 minimal set with a nonempty tail. For background to this problem, the reader is encouraged to consult Chapter 9 of [2] and the paper [1] by Day and Freese on Polin’s Variety. Loosely stated, Polin’s variety consists of algebras created by replacing the points of an (external) boolean algebra B by a family of (internal) boolean algebras {Ba : a ∈ B} so that whenever a ≥ b in B, there is a homomorphism ξ b from Ba to Bb. These homomorphisms are compatible in the sense that if a ≥ b ≥ c then ξ c = ξ b ◦ ξ c . The algebras come equipped with operations which allow one to recover both the internal and external boolean algebras. Our example is a simple modification of this idea wherein we replace the internal boolean structures by distributive lattices having a distinguished largest element 1. The proof that the resulting equational class satisfies some nontrivial congruence identity is practically the same as that presented by Day and Freese for Polin’s variety in [1]. The tame congruence theoretic properties of the class referred to earlier can easily be established. ∗1991 Mathematical Subject Classification. Primary 08B05; Secondary 08A30. †Support of NSERC is gratefully acknowledged