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...

Let S be a regular semigroup and C its lattice of congruences. We consider the sublattice Λ of C generated by σ-the least group, τ -the greatest idempotent pure, μ-the greatest idempotent separating and β-the least band congruence on S. To this end, we study the following special cases: (1) any three of these congruences generate a distributive lattice, (2) Λ is distributive, (3) the restrictio...

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...

The Lindenbaum algebra generated by the Abramsky ni-tary logic is a distributive lattice dual to an SFP-domain obtained as a solution of a recursive domain equation. We extend Abramsky's result by proving that the Lindenbaum algebra generated by the innnitary logic is a completely distributive lattice dual to the same SFP-domain. As a consequence soundness and completeness of the innnitary logi...

In works dealing with capacities (fuzzy measures) and the Choquet integral on finite spaces, it is usually considered that all subsets of the universe are measureable. Hence, all functions are integrable in the sense of Choquet. We consider the situation where some subsets are not measurable (not feasible), so that there are nonintegrable functions. Since this is a severe limitation in applicat...