As the main achievement of paper, we construct a three-generated, 2-distributive, atomless lattice that is not finitely presented. Also, paper contains following three observations. First, every coatomless three-generated has at least one atom. Second, give some sufficient conditions implying most atoms. Third, present meet-distributive with four

We characterize the finite intervals of the Muchnik lattice by proving that they form a certain proper subclass of the finite distributive lattices. We also discuss infinite intervals, mainly to conclude that much more is possible here than for the related Medvedev lattice.

An inequality between the number of coverings in the ordered set J(Con L) of join irreducible congruences on a lattice L and the size of L is given. Using this inequality it is shown that this ordered set can be computed in time O(n2 log2 n), where n = |L|. This paper is motivated by the problem of efficiently calculating and representing the congruence lattice Con L of a finite lattice L. Of c...

We study paths between maximal chains, or “flags,” in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the...

We describe G-sets whose congruences satisfy some natural lattice or multiplicative restrictions. In particular, we determine G-sets with distributive, arguesian, modular, upper or lower semimodular congruence lattice as well as congruence n-permutable G-sets for n = 2, 2.5, 3.

It is proved that every (universal) algebra A with distributive and permutable structure lattice is isomorphic with the algebra of all global sections with compact supports of a sheaf of homomorphic images of A over a topological space. This completely generalises the corresponding result of Klaus Keimel for /-rings. Introduction. The following note is a preliminary report on the investigations...

We discuss the possible structures for and mutual relationships between a finite distributive lattice L, a maximal sublattice M of L and the corresponding ‘remainder’ R = L\M with the aid of Birkhoff duality, and contrast the results with the analogous situations for a general finite lattice L. @ 1999 Elsevier Science B.V. All rights reserved

We study some distributive lattices arising in the combinatorics of lattice paths. In particular, for the Dyck, Motzkin and Schröder lattices we describe the spectrum and we determine explicitly the Euler characteristic in terms of natural parameters of lattice paths. AMS Classification: Primary: 06D05; Secondary: 06A07.

We introduce the class of bounded distributive lattices with two operators, and ∇, the first between the lattice and the set of its ideals, and the second between the lattice and the set of its filters. The results presented can be understood as a generalization of the results obtained by S. Celani.

In this paper we present a method (linear in the size of the formal context) to solve the seriation problem for formal contexts. We show that any maximal solution can be represented by a PQ-Tree. Moreover, the set of PQ-Trees can be seen as a distributive lattice. This lattice yields a consensus method which deals with the multiple solutions.