2-normalization of lattices

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

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Abstract

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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 $v\left(t\right)\ge 0$. 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 ${N}_{k}\left(V\right)$ 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 ${N}_{k}\left(V\right)$ 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 ${N}_{2}\left(V\right)$ where $V$ is the type $\left(2,2\right)$ 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 ${N}_{2}\left(L\right)$ is precisely the class of all $3$-level inflations of lattices. We also produce a finite equational basis for the variety ${N}_{2}\left(L\right)$.

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