In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving ex...

Soft set theory was introduced by Molodtsov in 1999 as a general mathematical tool for dealing with problems that contain uncertainty. In this paper, we define concept of fuzzy soft lattice, fuzzy soft sublattice, complete fuzzy soft lattice, modular fuzzy soft lattice, distributive fuzzy soft lattice, fuzzy soft chain and study their basic properties.

The normal, or Dedekind-MacNeille, completion δ(L) of a distributive lattice L need not be distributive. However, δ(L) does contain a largest distributive sublattice β(L) containing L, and δ(L) is distributive if and only if β(L) is complete if and only if δ(L) = β(L). In light of these facts, it may come as a surprise to learn that β(L) was developed (in [1]) for reasons having nothing to do w...

Let L ∗ M denote the coproduct of the bounded distributive lattices L and M . At the 1981 Banff Conference on Ordered Sets, the following question was posed: What is the largest class L of finite distributive lattices such that, for every non-trivial Boolean lattice B and every L ∈ L, B ∗ L = B ∗ L′ implies L = L′? In this note, the problem is solved.

Every lattice is isomorphic to a lattice whose elements are sets of sets and whose operations are intersection and the operation ∨∗ defined by A ∨∗ B = A ∪ B ∪ {Z : (∃X ∈ A)(∃Y ∈ B)X ∩ Y ⊆ Z}. This representation spells out precisely Birkhoff’s and Frink’s representation of arbitrary lattices, which is related to Stone’s set-theoretic representation of distributive lattices. (AMS Subject Classi...

A distributive lattice L with 0 is finitary if every interval is finite. A function f : N0 N0 is a cover function for L if every element with n lower covers has f (n) upper covers. In this paper, all finitary distributive lattices with non-decreasing cover functions are characterized. A 1975 conjecture of Richard P. Stanley is thereby settled. 2000 Academic Press

By studying two unsharp quantum structures, namely extended lattice ordered effect algebras and lattice ordered QMV algebras, we obtain some characteristic theorems of MV algebras. We go on to discuss automata theory based on these two unsharp quantum structures. In particular, we prove that an extended lattice ordered effect algebra (or a lattice ordered QMV algebra) is an MV algebra if and on...