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...

The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not...

A finite distributive latticeD can be representedas the congruence lattice, ConL, of a finite lattice L. We shall discuss the combinatorial aspects of such—and related—representations, specifically, optimal size, breadth, and degree of symmetry.