2-Normalization of lattices

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).

k-Normalization and (k + 1)-level inflation of varieties

“…There are many classes of varieties which satisfy the conditions of Theorem 3.2 or Corollary 3.4. These include any variety which satisfies an idempotent identity, or even a consequence of idempotence such as x ≈ t(x) for a term t of depth at least k. As a special case, we see that the construction from [1] for 2-normalizations of lattices did not actually need the order-theoretic property of lattices, but only the idempotence of the meet and join operations. However, there are many varieties for which there is no term t fulfilling conditions (C1) and (C2).…”

“…They showed that any algebra in N k (V ) is a homomorphic image of a k-choice algebra constructed from an algebra in V . In [1], Chajda, Cheng and Wismath also studied the algebras of the variety N 2 (L), the 2-normalization of the variety L of all lattices. Using the order-theoretic nature of lattices, they introduced a modification of 2-choice algebras called the 3-level inflation of an algebra, and showed that the variety N 2 (L) consists exactly of all 3-level inflations of lattices.…”