Let ~ H and ~ K be finite composition series of length h in a group G. The intersections of their members form a lattice CSL( ~ H, ~ K) under set inclusion. Our main result determines the number N(h) of (isomorphism classes) of these lattices recursively. We also show that this number is asymptotically h!/2. If the members of ~ H and ~ K are considered constants, then there are exactly h! such ...

A topology ^"i is said to cover another topology T3 if ^Ç^i and no other topology may be included between the two. In this paper, we characterize the relationship between a Tj-topology and its covers. This characterization is used to prove that the lattice of 7\-topologies is both upper and lower semimodular. We also prove that the sublattice generated by the covers of a 7\-topology is isomorph...

We prove that every finite distributive lattice D can be represented as the congruence lattice of a rectangular lattice K in which all congruences are principal. We verify this result in a stronger form as an extension theorem.

We show that the subgroup lattice of any finite group satisfies Frankl’s UnionClosed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.

The aimof this paper is to study local two-dimensional languages froman algebraic point of view.We show that local two-dimensional languages over a finite alphabet, with the usual relation of set inclusion, form a lattice. The simplest case Loc1 of local languages defined over the alphabet consisting of one element yields a distributive lattice, which can be easily described. In the general cas...

We introduce the concept of a bounded below set in a lattice. This can be used to give a generalization of Rota's broken circuit theorem to any finite lattice. We then show how this result can be used to compute and combinatorially explain the Mo bius function in various examples including non-crossing set partitions, shuffle posets, and integer partitions in dominance order. Next we present a ...