2-Normalization of lattices

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). This research was supported by Research Project MSM6198959214 of the Czech Government and by NSERC of Canada.