# 2-normalization of lattices

Ivan Chajda; W. Cheng; S. L. Wismath

Czechoslovak Mathematical Journal (2008)

- Volume: 58, Issue: 3, page 577-593
- ISSN: 0011-4642

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topChajda, Ivan, Cheng, W., and Wismath, S. L.. "2-normalization of lattices." Czechoslovak Mathematical Journal 58.3 (2008): 577-593. <http://eudml.org/doc/37854>.

@article{Chajda2008,

abstract = {Let $\tau $ be a type of algebras. A valuation of terms of type $\tau $ is a function $v$ assigning to each term $t$ of type $\tau $ a value $v(t) \ge 0$. For $k \ge 1$, an identity $s \approx t$ of type $\tau $ is said to be $k$-normal (with respect to valuation $v$) if either $s = t$ or both $s$ and $t$ have value $\ge k$. Taking $k = 1$ with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called $k$-normal (with respect to the valuation $v$) if all its identities are $k$-normal. For any variety $V$, there is a least $k$-normal variety $N_k(V)$ containing $V$, namely 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 their paper (Algebra Univers., 50, 2003, pp.107-128) and an algebraic characterization of the elements of $N_k(V)$ in terms of the algebras in $V$ was given in (Algebra Univers., 51, 2004, pp. 395–409). In this paper we study the algebras of the variety $N_2(V)$ where $V$ is the type $(2,2)$ variety $L$ of lattices and our valuation is the usual depth valuation of terms. We introduce a construction called the $3$-level inflation of a lattice, and use the order-theoretic properties of lattices to show that the variety $N_2(L)$ is precisely the class of all $3$-level inflations of lattices. We also produce a finite equational basis for the variety $N_2(L)$.},

author = {Chajda, Ivan, Cheng, W., Wismath, S. L.},

journal = {Czechoslovak Mathematical Journal},

keywords = {2-normal identities; lattices; 2-normalized lattice; 3-level inflation of a lattice; 2-normal identities; lattices; 2-normalized lattice; 3-level inflation of a lattice},

language = {eng},

number = {3},

pages = {577-593},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {2-normalization of lattices},

url = {http://eudml.org/doc/37854},

volume = {58},

year = {2008},

}

TY - JOUR

AU - Chajda, Ivan

AU - Cheng, W.

AU - Wismath, S. L.

TI - 2-normalization of lattices

JO - Czechoslovak Mathematical Journal

PY - 2008

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 58

IS - 3

SP - 577

EP - 593

AB - Let $\tau $ be a type of algebras. A valuation of terms of type $\tau $ is a function $v$ assigning to each term $t$ of type $\tau $ a value $v(t) \ge 0$. For $k \ge 1$, an identity $s \approx t$ of type $\tau $ is said to be $k$-normal (with respect to valuation $v$) if either $s = t$ or both $s$ and $t$ have value $\ge k$. Taking $k = 1$ with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called $k$-normal (with respect to the valuation $v$) if all its identities are $k$-normal. For any variety $V$, there is a least $k$-normal variety $N_k(V)$ containing $V$, namely 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 their paper (Algebra Univers., 50, 2003, pp.107-128) and an algebraic characterization of the elements of $N_k(V)$ in terms of the algebras in $V$ was given in (Algebra Univers., 51, 2004, pp. 395–409). In this paper we study the algebras of the variety $N_2(V)$ where $V$ is the type $(2,2)$ variety $L$ of lattices and our valuation is the usual depth valuation of terms. We introduce a construction called the $3$-level inflation of a lattice, and use the order-theoretic properties of lattices to show that the variety $N_2(L)$ is precisely the class of all $3$-level inflations of lattices. We also produce a finite equational basis for the variety $N_2(L)$.

LA - eng

KW - 2-normal identities; lattices; 2-normalized lattice; 3-level inflation of a lattice; 2-normal identities; lattices; 2-normalized lattice; 3-level inflation of a lattice

UR - http://eudml.org/doc/37854

ER -

## References

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- Denecke, K., Wismath, S. L., 10.1007/s00012-004-1864-2, Algebra Univers. 51 (2004), 395-409. (2004) Zbl1080.08002MR2082134DOI10.1007/s00012-004-1864-2
- Denecke, K., Wismath, S. L., 10.1007/s00012-003-1824-2, Algebra Univers. 50 (2003), 107-128. (2003) Zbl1092.08003MR2026831DOI10.1007/s00012-003-1824-2
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