Computing congruence lattices of finite lattices

An inequality between the number of coverings in the ordered set of join irreducible congruences on a lattice and the size of is given. Using this inequality it is shown that this ordered set can be computed in time , where .

     

On finite and totally finite elements in vector lattices

, X , , , M l- M X , ( M) M .     

On finite lattices which are embeddable in subsemigroup lattices

Communicated by L. N. Shevrin     

Koszul filtrations and finite lattices

In this note, we characterize when a finite lattice is distributive in terms of the existences of some particular classes of Koszul filtrations.     

On finite critical lattices

Critical lattices are considered, i.e., lattices without nontrivial endomorphisms and not containing nontrivial proper sublattices without nontrivial endomorphisms. It is proved that there exist n-element critical sublattices for any n ≥ 21.     

Fixed points and complements in finite lattices

Let L be a finite lattice and . Suppose that B is a set of lower semicomplements of x, which includes all complements of x. We show that the partially ordered set has the fixed point property.     

Congruences of fork extensions of slim,planar, semimodular lattices

For a slim, planar, semimodular lattice L and a covering square S of L, G. Czédli and E. T. Schmidt introduced the fork extension, L[S], which is also a slim, planar, semimodular lattice. This paper investigates when a congruence of L extends to L[S].     

Semimodularity in lattices of finite length

For a lattice L of finite length we denote by J(L) the set of all join-irreducible elements (≠0) of L. By u′ we mean the uniquely determined lower cover of an element u?J(L). Our main result is the following theorem: A lattice L of finite length is (upper) semimodular if and only if it satisfies the exchange property (EP): c?b∨u and $\text{c?b∨u′}$ imply u?b∨c∨u′ (b, c?L;u?J(L)).     

Congruence-preserving extensions of finite lattices to sectionally complemented lattices

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 extension to a finite sectionally complemented lattice.

     

Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.     

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.     

Distance characterizations of finite lattices

Semimodular, modular and distributive finite lattices are characterized by means of convex sublattices and distance closed sets of the Hasse diagram graphs.     

Coordinatization of finite join-distributive lattices

Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as lattices with unique irreducible decompositions, introduced by R.P. Dilworth in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since then. Let L be a join-distributive lattice of length n, and let k denote the width of the set of join-irreducible elements of L. We prove that there exist k ? 1 permutations acting on {1, . . . , n} such that the elements of L are coordinatized by k-tuples over {0, . . . , n}, and the permutations determine which k-tuples are allowed. Since the concept of join-distributive lattices is equivalent to that of antimatroids and convex geometries, our result offers a coordinatization for these combinatorial structures.