In this paper, the concepts of $L$-concave structures, concave $L$-interior operators and concave $L$-neighborhood systems are introduced. It is shown that the category of $L$-concave spaces and the category of concave $L$-interior spaces are isomorphic, and they are both isomorphic to the category of concave $L$-neighborhood systems whenever $L$ is a completely distributive lattice. Also, it i...

By a lattice we shall always mean a distributive lattice which is bounded, i.e. has both a bottom element 0 and a top element 1. Lattice homomorphisms will always be assumed to preserve 0 and 1. By a modality on a (distributive) lattice L = (L, ∧, ∨, ≤, 0, 1) is meant a map : L → L satisfying (1) 1 = 1, (2) (x ∧ y) = x ∧ y for x, y ∈ L. The pair (L, ) will be called a modalized (distributive) l...

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...

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...

the aim of this paper is to establish a fuzzy version of the dualitybetween domains and completely distributive lattices. all values aretaken in a fixed frame $l$. a definition of (strongly) completelydistributive $l$-ordered sets is introduced. the main result inthis paper is that the category of fuzzy domains is dually equivalentto the category of strongly completely distributive $l$-ordereds...

A (distributive) p-algebra is an algebra 〈L;∨,∧, ∗, 0, 1〉 whose reduct 〈L;∨,∧, 0, 1〉 is a bounded (distributive) lattice and whose unary operation ∗ is characterized by x ≤ a if and only if a ∧ x = 0. If L is a p-algebra, B(L) = { x ∈ L : x = x } and D(L) = { x ∈ L : x = 1 } then 〈B(L);∪,∧, 0, 1〉 is a Boolean algebra when a ∪ b is defined to be (a∗ ∧ b∗)∗, for any a, b ∈ B(L), D∗(L) = { x ∨ x∗ ...

The aim of this paper is to establish a fuzzy version of the dualitybetween domains and completely distributive lattices. All values aretaken in a fixed frame $L$. A definition of (strongly) completelydistributive $L$-ordered sets is introduced. The main result inthis paper is that the category of fuzzy domains is dually equivalentto the category of strongly completely distributive $L$-ordereds...

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.

In the early eighties, A. Huhn proved that if D, E are finite distributive lattices and ψ : D → E is a {0}-preserving join-embedding, then there are finite lattices K, L and there is a lattice homomorphism φ : K → L such that ConK (the congruence lattice of K) is isomorphic to D, ConL (the congruence lattice of L) is isomorphic to E, and the natural induced mapping extφ : ConK → ConL represents...

Let L be a distributive lattice and R(L) the associated Hibi ring. We compute regR(L) when L is a planar lattice and give bounds for regR(L) when L is nonplanar, in terms of the combinatorial data of L. As a consequence, we characterize the distributive lattices L for which the associated Hibi ring has a linear resolution.