We prove that every distributive algebraic lattice with at most א1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The א1 bound is optimal, as we find a distributive algebraic lattice D with א2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence n...

We prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

In this paper, we define the concepts of fuzzy initial and final segments in an Almost Distributive Lattice (ADL) certain properties these are discussed. It is proved that set forms a complete lattice ADL A if only bounded distributive lattice.

Let D be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite distributive lattice D, the set {d, d} is definable, where d and d are the isomorphism types of D and its opposite (D turned upside down). We prove that the only non-identit...

We prove that every distributive algebraic lattice with at most א1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The א1 bound is optimal, as we find a distributive algebraic lattice D with א2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence n...

In this note, we show that any distributive lattice is isomorphic to the set of reachable configurations of an Edge Firing Game. Together with the result of James Propp, saying that the set of reachable configurations of any Edge Firing Game is always a distributive lattice, this shows that the two concepts are equivalent.

This paper constructs a logic of soft constraints where the set of degrees of preference forms a partially ordered set. When the partially ordered set is a distributive lattice, this reduces to the idempotent semiring-based CSP approach, and the lattice operations can be used to define a sound and complete proof theory. For the general case, it is shown how sound and complete deduction can be p...

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∗ ...