Minimal Characteristic Algebras for Rectangular k-Normal Identities

Abstract

A characteristic algebra for a hereditary property of identities of a fixed type τ is an algebra [Formula: see text] such that for any variety V of type τ, we have [Formula: see text] if and only if every identity satisfied by V has the property p. This is equivalent to [Formula: see text] being a generator for the variety determined by all identities of type τ which have property p. Płonka has produced minimal (smallest cardinality) characteristic algebras for a number of hereditary properties, including regularity, normality, uniformity, biregularity, right- and leftmost, outermost, and external-compatibility. In this paper, we use a construction of Płonka to study minimal characteristic algebras for the property of rectangular k-normality. In particular, we construct minimal characteristic algebras of type (2) for k-normality and rectangularity for 1 ≤ k ≤ 3.