# 2-Normalization of lattices

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## Abstract

Let *τ* be a type of algebras. A valuation of terms of type *τ* is a function *v* assigning to each term *t* of type *τ* a value *v*(*t*) ⩾ 0. For *k* ⩾ 1, an identity *s* ≈ *t* of type *τ* is said to be *k*-normal (with respect to valuation *v*) if either *s* = *t* or both *s* and *t* have value ⩾ *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*).

## Keywords

2-normal identities lattices 2-normalized lattice 3-level inflation of a lattice## Preview

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