A lattice L is said to satisfy (the lattice theoretic version of) Frankl’s conjecture if there is a join-irreducible element f ∈ L such that at most half of the elements x of L satisfy f ≤ x. Frankl’s conjecture, also called as union-closed sets conjecture, is well-known in combinatorics, and it is equivalent to the statement that every finite lattice satisfies Frankl’s conjecture. Let m denote...

We prove that every finite lattice has a congruence-preserving extension to a finite semimodular lattice.

In a ranked lattice, we consider two maximal chains, or “flags” to be i-adjacent if they are equal except possibly on rank i . Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a “Jordan-Hölder permutation” between any two flags. This permutation has the properties of an Sn-distance function on the chamber system of flags. Using these noti...

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

The lattice of closed subsets of a set under such a closure operator is semimodular. Perhaps the best known example of a closure operator satisfying the exchange principle is the closure operator on a vector space W where for X ___ W we let C(X) equal the span of X. The lattice of C-closed subsets of W is isomorphic to Con(W) in a natural way; indeed, if Y _~ W x W and Cg(Y) denotes the congrue...

This paper deals with the relations between graph automorphisms and direct factors of a semimodular lattice of locally finite length.

In this paper we prove that if !.l' is a finite lattice. and r is an integral valued function on !.l' satisfying some very natural then there exists a finite geometric (that is.• semimodular and atomistic) lattice containing asa sublatticesuch that r !.l'restricted to Sf. Moreover. we show that if, for all intervals of. semimodular lattices of length at most r(e) are given. then can be chosen t...

A recent result of G. Czédli and E. T. Schmidt gives a construction of slim (planar) semimodular lattices from planar distributive lattices by adding elements, adding “forks”. We give a construction that accomplishes the same by deleting elements, by “resections”.

We present a new gluing construction for semimodular lattices, related to the Hall–Dilworth construction. The gluing constructions in the lattice theory started with a paper of M. Hall and R. P. Dilworth [4] to prove that there exists a modular lattice that cannot be embedded in any complemented modular lattice. This construction is the following: let K and L be lattices, let F be a filter of K...

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.