2-normalization of lattices

Title:

2-normalization of lattices

Creator:

Chajda, Ivan, Cheng, W., and Wismath, S. L.

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:3e173642-9483-4f2d-b5f3-e0ca7012bd73
uuid:3e173642-9483-4f2d-b5f3-e0ca7012bd73

Subject:

2-normal identities, lattices, 2-normalized lattice, and 3-level inflation of a lattice

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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 $v(t) \geq 0$. For $k \geq 1$, 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 $\geq k$. 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 {\it $3$-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 $N_2(L)$.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

577-593

Source:

Czechoslovak Mathematical Journal | 2008 Volume:58 | Number:3

Harvested from:

CDK

Metadata only:

false