Fregean subtractive varieties with definable congruence

In this paper we investigate subtractive varieties of algebras that are Fregean in order to get structure theorems about them. For instance it turns out that a subtractive variety is Fregean and has equationally definable principal congruences if and only if it is termwise equivalent to a variety of Hilbert algebras with compatible operations. Several examples are provided to illustrate the theory.

Double MSn-algebras and double Kn.m-algebras

The variety O2 of all algebras (L; ∧, ∨, f, g, 0, 1) of type (2, 2, 1, 1, 0, 0) such that (L; ∧, ∨, f, 0, 1) and (L; ∧, ∨, g, 0, 1) are Ockham algebras is introduced, and, for n, m εℕ, its subvarieties DMSn, of double MSn-algebras, and DKn,m, of double Kn,m-algebras, are considered. It is shown that DKn,m has equationally definable principal congruences: a description of principal congruences on double Kn,m-algebras is given and simplified for double MSn-algebras. A topological duality for O2-algebras is developed and used to determine the subdirectly irreducible algebras in DKn,m and in DMSn. Finally, MSn-algebras which are reduct of a (unique) double MSn-algebra are characterized.

Varieties generated by finite BCK-algebras

Iséki's BCK-algebras form a quasivariety of groupoids and a finite BCK-algebra must satisfy the identity (En): xyn = xyn+1, for a suitable positive integer n. The class of BCK-algebras which satisfy (En) is a variety which has strongly equationally definable principal congruences, congruence-3-distributivity, and congruence-3-permutability. Thus, a finite BCK-algebra generates a 3-based variety of BCK-algebras. The variety of bounded commutative BCK-algebras which satisfy (En) is generated by n finite algebras, each of which is semiprimal.