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

Valerie Cheng, and Shelly Wismath. "k-Normalization and (k+1)-level inflation of varieties." Discussiones Mathematicae - General Algebra and Applications 28.1 (2008): 49-62. <http://eudml.org/doc/276873>.

@article{ValerieCheng2008,
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 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.},
author = {Valerie Cheng, Shelly Wismath},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {k-normal identities; k-normalization of a variety; (k+1)-level inflation of algebras; -normal; variety; -normalization; depth; -level inflation; lattice},
language = {eng},
number = {1},
pages = {49-62},
title = {k-Normalization and (k+1)-level inflation of varieties},
url = {http://eudml.org/doc/276873},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Valerie Cheng
AU - Shelly Wismath
TI - k-Normalization and (k+1)-level inflation of varieties
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2008
VL - 28
IS - 1
SP - 49
EP - 62
AB - 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.
LA - eng
KW - k-normal identities; k-normalization of a variety; (k+1)-level inflation of algebras; -normal; variety; -normalization; depth; -level inflation; lattice
UR - http://eudml.org/doc/276873
ER -