In this paper the authors have studied the Glivenko congruence R in a 0-distributive nearlattice S defined by " () R b a ≡ if and only if 0 = ∧ x a is equivalent to 0 = ∧ x b for each S x ∈ ". They have shown that the quotient nearlattice R S is weakly complemented. Moreover, R S is distributive if and only if S is 0-distributive. They also proved that every Sectionally complemented nearlattice...

From a well-known decomposition theorem, we propose a tree representation for distributive and simplicial lattices. We show how this representation (called ideal tree) can be efficiently computed (linear time in the size of the lattice given by any graph whose transitive closure is the lattice) and compared with respect to time and space complexity. As far as time complexity is concerned, we si...

In the early forties, R. P. Dilworth proved his famous result: Every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In one of our early papers, we presented the first published proof of this result; in fact we proved: Every finite distributive lattice D can be represented as the congruence lattice of a finite sectionally complemented lattice L....

If R is a commutative ring with identity and ≤ is defined by letting a ≤ b mean ab = a or a = b, then (R,≤) is a partially ordered ring. Necessary and sufficient conditions on R are given for (R,≤) to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings Zn of integers mod n for n ≥ 2. In particular, if R is reduced, then (R,≤) is a la...

We give a new characterization of sober spaces in terms of their completely distributive lattice of saturated sets. This characterization is used to extend Abramsky’s results about a domain logic for transition systems. The Lindenbaum algebra generated by the Abramsky finitary logic is a distributive lattice dual to an SFP-domain obtained as a solution of a recursive domain equation. We prove t...

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on ∆-bonds with inva...

It is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1, Y2 such that Y1 c X1, Y2 c X2 and Y1 U Y2 = X1 U X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all sep...

Let W be a Coxeter group. We define an element w ~ W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose membe...

A net (xγ)γ∈Γ in a locally solid Riesz space (X,τ) is said to be unbounded τ-convergent x if |xγ−x|∧u⟶τ0 for all u∈X+. We recall that there linear topology uτ on X such τ-convergence coincides with uτ-convergence. It turns out characterised as the weakest which τ order bounded subsets. this motivation we introduce, uniform lattice (L,u), uniformity u⁎ L u subsets of L. shown induced by (X,τ), t...