Ágnes Szendrei

Equivalence of operations with respect to discriminator clones

For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations.

CLONES OF ALGEBRAS WITH PARALLELOGRAM TERMS

We describe a manageable set of relations that generates the finitary relational clone of an algebra with a parallelogram term. This result applies to any algebra with a Maltsev term and to any algebra with a near unanimity term. One consequence of the main result is that on any finite set and for any finite k there are only finitely many clones of algebras with a k-ary parallelogram term which generate residually small varieties.

The set of types of a finitely generated variety

Abstract The paper presents an algorithm of polynomial time complexity to compute the type set of a finite algebraic system A , as defined in the monograph of McKenzie and Hobby: ‘The Structure of Finite Algebras’. To do so, it introduces the concept of a subtrace, and uses subtraces to characterize the type set of A . It is also shown that to calculate the type set of the variety generated by A is more difficult, by presenting various examples, in which a given type occurs only in subalgebras of high powers of A .

Self-Rectangulating Varieties of Type 5

We show that a locally finite variety which omits abelian types is self-rectangulating if and only if it has a compatible semilattice term operation. Such varieties must have type-set {5}. These varieties are residually small and, when they are finitely generated, they have definable principal congruences. We show that idempotent varieties with a compatible semilattice term operation have the congruence extension property.

Commuting Polynomial Operations of Distributive Lattices

We describe which pairs of distributive lattice polynomial operations commute.

Clones with finitely many relative R-classes

For each clone

GROWTH RATES OF ALGEBRAS, I: POINTED CUBE TERMS

{\mathcal {C}}

A CLASSIFICATION OF STRICTLY SIMPLE ALGEBRAS WITH TRIVIAL SUBALGEBRAS

on a set A there is an associated equivalence relation analogous to Green’s