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 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 in terms of the algebras in V was given in [4]. In [1] a simplified version of this characterization of 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1134,
author = {Valerie Cheng and Shelly Wismath},
title = {k-Normalization and (k+1)-level inflation of varieties},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {28},
year = {2008},
pages = {49-62},
zbl = {1156.08001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1134}
}
Valerie Cheng; Shelly Wismath. k-Normalization and (k+1)-level inflation of varieties. Discussiones Mathematicae - General Algebra and Applications, Tome 28 (2008) pp. 49-62. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1134/
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