A nearlattice is a meet semilattice A in which every initial segment Ap := {x : x ≤ p} happens to be a join semilattice (hence, a lattice) with respect to the natural ordering of A. If all lattices Ap are distributive, the nearlattice itself is said to be distributive. It is known that a distributive nearlattice can be represented as a nearlattice of sets. We call an algebra (A,∧,∨) of type (2,...

A semimodular lattice L of finite length will be called an almost-geometric lattice, if the order J(L) of its nonzero join-irreducible elements is a cardinal sum of at most two-element chains. We prove that each finite distributive lattice is isomorphic to the lattice of congruences of a finite almost-geometric lattice.

Stone algebras have been characterized by Chen and Grätzer in terms of triples (B;D;'), whereD is a distributive lattice with 1; B is a Boolean algebra, and ' is a bounded lattice homomorphism from B into the lattice of …lters of D: If D is bounded, the construction of these characterizing triples is much simpler, since the homomorphism ' can be replaced by one from B into D itself. The triple ...

We prove that the skeleton of a product of finitely many finite distributive lattices is isomorphic to the product of skeletons of its factors. Thus, it is possible to construct finite distributive lattices with a given directly reducible skeleton by reducing the problem to the skeleton factors. Although not all possible lattices can be obtained this way, we show that it works for the smallest ...

The concept of extended ideals in an Almost distributive Lattice is introduced and studied their properties.

We show that a finite distributive lattice can be embedded into the r.e. degrees preserving least and greatest element if and only if the lattice contains a join-irreducible noncappable element.

In this paper, we verify the diameter of zero divisor graphs with respect to direct product. Keywords—Atomic lattice, complement of graph, diameter, direct product of lattices, 0-distributive lattice, girth, product of graphs, prime ideal, zero divisor graph.

The main goal of this paper is to study the lattice of T L-submodules of a module. It is well-known that the lattice of submodules of a module is modular. In this study, we prove an analogous result for L-sets that is the lattice of L-submodules of a module is modular for an infinitely ∨-distributive lattice.

We construct an algebraic distributive lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R. P. Dilworth, from the forties. The lattice D has compact top element and אω+1 compact elements. Our results extend to any algebra possessing a polynomially definable structure of a joinsemilattice with largest ...

We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finite Boolean 〈∨, 0〉-semilattices with 〈∨, 0〉-embeddings, can be lifted with respect to the Conc functor on lattices, then so can every diagra...