A generalization of completely distributive lattices

Presented by Bjarni Jónsson.     

A Cayley theorem for distributive lattices

A Cayley-like representation theorem for distributive lattices is proved. Support of the research of the first author by the Czech Government Research Project MSM 6198959214 is gratefully acknowledged.     

Endoprimal distributive lattices

Presented by A. Szendrei.     

A size-width inequality for distributive lattices

We show that every collection ofw sets such that none contains any other generates at least 3w-2 sets under the operations of taking intersections and unions. In particular, we prove that if the finite distributive lattice ? contains an antichain of sizew, then |?| ≧3w, forw≠1, 2, 3, 6, where the minimal exceptional cases arise from the Boolean algebras ?_n withn=0, 1, 2, 3, 4 atoms.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

Finite distributive lattices are congruence lattices of almost-geometric lattices

A semimodular lattice L of finite length will be called an almost-geometric lattice if the order J(L) of its nonzero join-irreducible elements is a cardinal sum of at most two-element chains. We prove that each finite distributive lattice is isomorphic to the lattice of congruences of a finite almost-geometric lattice.     

A topological characterization of complete distributive lattices

An ordered compact space is a compact topological space X, endowed with a partially ordered relation, whose graph is a closed set of X × X (cf. [4]). An important subclass of these spaces is that of Priestley spaces, characterized by the following property: for every x, y ? X with x ? y there is an increasing clopen set A (i.e. A is closed and open and such that a ? A, a ? z implies that z?A) which separates x from y, i.e., x ? A and y ? A. It is known (cf. [5, 6]) that there is a dual equivalence between the category Ld01 of distributive lattices with least and greatest element and the category P of Priestley spaces.In this paper we shall prove that a lattice L ? Ld01 is complete if and only if the associated Priestley space X verifies the condition: (E0) D ? X, D is increasing and open implies $\text{D}{}^1$ is increasing clopen (where A¹ denotes the least increasing set which includes A).This result generalizes a well-known characterization of complete Boolean algebras in terms of associated Stone spaces (see [2, Ch. III, Section 4, Lemma 1], for instance).We shall also prove that an ordered compact space that fulfils (E0) is necessarily a Priestley space.     

Complete congruence lattices of join-infinite distributive lattices

Received July 26, 1993; accepted in final form July 16, 1996.     

Affine complete distributive lattices

We prove a characterization theorem for affine complete distributive lattices. To do so we introduce the notions of almost principal ideal and almost principal.This research was supported by GA SAV Grant 362/93.