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

Cheng, Valerie ; Wismath, Shelly

Discussiones Mathematicae. General Algebra and Applications, Tome 28 (2008) no. 1, pp. 49-62

Résumé

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 well-known 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.

Détail

Keywords: k-normal identities, k-normalization of a variety, (k+1)-level inflation of algebras