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

Discussiones Mathematicae - General Algebra and Applications (2008)

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

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topValerie 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

top- [1] I. Chajda, V. Cheng and S.L. Wismath, 2-Normalization of lattices, to appear in Czechoslovak Mathematics Journal. Zbl1174.08003
- [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] G.T. Clarke, Semigroup varieties of inflations of unions of groups, Semigroup Forum 23 (4) (1981), 311-319. Zbl0486.20033
- [4] K. Denecke and S.L. Wismath, A characterization of k-normal varieties, Algebra Universalis 51 (4) (2004), 395-409. Zbl1080.08002
- [5] K. Denecke and S.L. Wismath, Valuations of terms, Algebra Universalis 50 (1) (2003), 107-128. Zbl1092.08003
- [6] E. Graczyńska, On normal and regular identities, Algebra Universalis 27 (3) (1990), 387-397. Zbl0713.08007
- [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.