In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of zonotopes, ordered with a simple and classical dynamics, is the disj...

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff’s Distributive Lattices. Our FTFSDL is the form “A poset L finite semidistributive lattice if and only there exists set with some additional structure, such that isomorphic to admissible subsets ordered by inclusion; in this case, its structure are uniquely determined L.” The combinatorial abstract...

In this paper, countable compactness and the Lindel¨of propertyare defined for L-fuzzy sets, where L is a complete de Morgan algebra. Theydon’t rely on the structure of the basis lattice L and no distributivity is requiredin L. A fuzzy compact L-set is countably compact and has the Lindel¨ofproperty. An L-set having the Lindel¨of property is countably compact if andonly if it is fuzzy compact. ...

Let 〈D,≤〉 be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We characterize the order ideals in 〈D,≤〉 that are well-quasi-ordered by embeddability, and thus characterize the members of D that belong to at least one infinite anti-chain in D.

Every N „-categorical distributive lattice of finite breadth has a finitely axiomatizablc theory. This result extends the analogous result for partially ordered sets of finite width. 0. Introduction. This note is mainly concerned with the following theorem. Theorem 1. Every S „-categorical distributive lattice of finite breadth has a finitely axiomatizable theory. For general references on dist...

In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice, whenever the partially ordered set of the equivalence classes of R∩R−1 does not contain infinite ascending chains. We show that on this lattice can be defined three different kinds of complementation operations, and we...

In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice, whenever the partially ordered set of the equivalence classes of R∩R does not contain infinite ascending chains. We show that on this lattice can be defined three different kinds of complementation operations, and we d...

The least element 0 of a finite meet semi-distributive lattice is a meet of meet-prime elements. We investigate conditions under which the least element of an algebraic, meet semi-distributive lattice is a (complete) meet of meet-prime elements. For example, this is true if the lattice has only countably many compact elements, or if |L| < 2א0 , or if L is in the variety generated by a finite me...

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a 2-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not b...

There are several standard constructions by which an arbitrary ordered set P is extended to one in which every subset has an. infimum and a supremuma complete ordered set. The collection I(P) of all initial segments of P ordered by inclusion is one. [A subset I of P is an initial segment if, for x, y E P, x E I whenever x s y and y E I.] I(P) is a complete distributive lattice. AnoWr is the fam...