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

• Volume: 28, Issue: 1, page 49-62
• ISSN: 1509-9415

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## Abstract

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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 ${N}_{k}\left(V\right)$ 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 ${N}_{k}\left(V\right)$ in terms of the algebras in V was given in [4]. In [1] a simplified version of this characterization of ${N}_{k}\left(V\right)$ 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 ${N}_{k}\left(V\right)$ 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.

## How to cite

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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 -

## References

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1. [1] I. Chajda, V. Cheng and S.L. Wismath, 2-Normalization of lattices, to appear in Czechoslovak Mathematics Journal. Zbl1174.08003
2. [2] A. Christie, Q. Wang and S.L. Wismath, Minimal characteristic algebras for k-normality, Scientiae Mathematicae Japonicae 61 (3) (2005), 547-565. Zbl1080.08001
3. [3] G.T. Clarke, Semigroup varieties of inflations of unions of groups, Semigroup Forum 23 (4) (1981), 311-319. Zbl0486.20033
4. [4] K. Denecke and S.L. Wismath, A characterization of k-normal varieties, Algebra Universalis 51 (4) (2004), 395-409. Zbl1080.08002
5. [5] K. Denecke and S.L. Wismath, Valuations of terms, Algebra Universalis 50 (1) (2003), 107-128. Zbl1092.08003
6. [6] E. Graczyńska, On normal and regular identities, Algebra Universalis 27 (3) (1990), 387-397. Zbl0713.08007
7. [7] I.I. Mel'nik, Nilpotent shifts of varieties, (in Russian), Mat. Zametki, 14 (1973), 703-712; English translation in: Math. Notes 14 (1973), 962-966.

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