If M is a finite complemented modular lattice with n atoms and D is a bounded distributive lattice, then the Priestley power M [D] is shown to be isomorphic to the poset of normal elements of Dn, thus solving a problem of E. T. Schmidt from 1974. It is shown that there exist a finite modular lattice A not having M4 as a sublattice and a finite modular lattice B such that A⊗B is not semimodular,...

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...

Let L1 be a finite lattice with an ideal L2. Then the restriction map is a {0, 1}-homomorphism from ConL1 into ConL2. In 1986, the present authors published the converse. If D1 and D2 are finite distributive lattices, and φ : D1 → D2 is a {0, 1}-homomorphism, then there are finite lattices L1 and L2 with an embedding η of L2 as an ideal of L1, and there are isomorphisms ε1 : ConL1 → D1 and ε2 :...

In 1990, we published the following result: Let m be a regular cardinal > א0. Every m-algebraic lattice L can be represented as the lattice of m-complete congruence relations of an m-complete modular lattice K. In this note, we present a short proof of this theorem. In fact, we present a significant improvement: The lattice K we construct is 2-distributive.

T. Katri n ak proved the following theorem: Every nite distributive lattice is the congruence lattice of a nite p-algebra. We provide a short proof, and a generalization, of this result.

Answering a question raised by Terwijn, we give a lattice-theoretic characterization of the intervals of the Muchnik lattice as a subset of the distributive lattices, valid for all intervals satisfying a cardinality constraint on the size of antichains.

In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving ex...

Soft set theory was introduced by Molodtsov in 1999 as a general mathematical tool for dealing with problems that contain uncertainty. In this paper, we define concept of fuzzy soft lattice, fuzzy soft sublattice, complete fuzzy soft lattice, modular fuzzy soft lattice, distributive fuzzy soft lattice, fuzzy soft chain and study their basic properties.

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...