Direct decompositions of non-algebraic complete lattices

ar X iv : m at h . G M / 0 50 13 73 v 1 2 2 Ja n 20 05 DIRECT DECOMPOSITIONS OF NON - ALGEBRAIC COMPLETE LATTICES

For a given complete lattice L, we investigate whether L can be decomposed as a direct product of directly indecomposable lattices. We prove that this is the case if every element of L is a join of join-irreducible elements and dually, thus extending to non-algebraic lattices a result of L. Libkin. We illustrate this by various examples and counterexamples.

Direct Decompositions of Atomistic Algebraic Lattices

Distributive lattices with strong endomorphism kernel property as direct sums

Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem  2.8}). We shall determine the structure of special elements (which are introduced after  Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of ...

Direct Product Decompositions of Infinitely Distributive Lattices

Let α be an infinite cardinal. Let Tα be the class of all lattices which are conditionally α-complete and infinitely distributive. We denote by T ′ σ the class of all lattices X such that X is infinitely distributive, σ-complete and has the least element. In this paper we deal with direct factors of lattices belonging to Tα. As an application, we prove a result of Cantor-Bernstein type for latt...

Direct Product Decompositions of Lattices , Closures andRelation

In this paper we study the direct product decompositions of closure operations and lattices of closed sets. We characterize the direct product decompositions of lattices of closed sets in terms of closure operations, and nd those decompositions of lattices which correspond to the decompositions of closures. If a closure on a nite set is represented by its implication base (i.e. a binary relatio...