k-Normalization and (k + 1)-level inflation of varieties

Abstract

Let τ be a type of algebras. A common measurement of the complexity of terms of type τ is the depth of a term. For k ≥ 1, an identity s ≈ t of type τ is said to be k-normal (with respect to this depth complexity measurement) if either s = t or both s and t have depth ≥ k. A variety is called k-normal if all its identities are k-normal. Taking k = 1 with respect to the usual depth valuation of terms gives the wellknown property of normality of identities or varieties. For any variety V , there is a least k-normal variety N k (V) containing V , the variety determined by the set of all k-normal identities of V. The concept of k-normalization was introduced by K. Denecke and S.L. Wismath in [5], and an algebraic characterization of the elements of N k (V) in terms of the algebras in V was given in [4]. In [1] a simplified version of this characterization of N k (V) was given, in the special case of the 2-normalization of the variety V of all lattices, using a construction called the 3-level inflation of a lattice. In this paper we show that the analogous (k + 1)-level inflation can be used to characterize the algebras of N k (V) for any variety V having a unary term which satisfies two technical conditions. This includes any variety V which satisfies x ≈ t(x) for some unary term t of depth at least k, and in particular any variety, such as the variety of lattices, which satisfies an idempotent identity.

Minimal Characteristic Algebras for Rectangular k-Normal Identities

“…Let k be a positive integer. In this section, we use the (k + 1)-level inflation construction from [1] to deduce several important properties that any characteristic algebra for rectangular k-normality must possess. First we state the key definition and the results we will need.…”

“…Lemma 6.4 and its proof in [1] tell us a great deal about the structure of any characteristic algebra for rectangular k-normality. This is the content of the next lemma.…”

“…Massé, Wang and Wismath [6] also used these to find minimal characteristic algebras for the intersection of k-normality and leftmost, using P lonka's construction, and they showed that this construction does not always produce a minimal characteristic algebra. Here we introduce some new techniques from [1] to determine whether the algebras produced by P lonka's construction are minimal or not for type (2) and k = 1, 2, 3.…”