The Bottom of the Lattice of BCK-varieties

It is evident from the definition that the class of all BCK-algebras is a quasivariety. Yet it is not a variety, nor does the largest subvariety of BCK exist—as shown in Wroński [2], Kabziński & Wroński [3], respectively. Several subvarieties of BCK have been isolated and thoroughly investigated, which has led to a substantial body of results (cf. e.g., Blok & Raftery [4]). Nevertheless, the question of describing the bottom of the lattice of BCK-varieties, raised in Pa lasiński & Wroński [5], has been remaining open. Let us now remind some basic concepts that will be of use in the sequel. An ideal I of a BCK-algebra A is a subset of A, such that: 0 ∈ I ; and whenever b ∈ I, a−̇b ∈ I , then a ∈ I as well. By a BCK-congruence of A we mean a congruence Φ such that A/Φ is a BCK-algebra. If Θ is any congruence of A, then its equivalence class of 0, [0]Θ, is an ideal of A. Conversely, for each ideal I of A there is a congruence Θ with [0]Θ = I . In general, this congruence need not be unique, and it need not be a BCK-congruence, either. However, the congruence ΘI , defined by: (a, b) ∈ ΘI iff a−̇b ∈ I and b−̇a ∈ I turns out to be the largest congruence with [0]Θ = I , and, at the same time, the unique BCK-congruence with this property.