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.

in this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. it is proved that an abelian group of finite exponent is strongly coatomic.

The class of Strongly Signable partially ordered sets is introduced and studied. It is show that strong signability, reminiscent of Björner–Wachs’ recursive coatom orderability, provides a useful and broad sufficient condition for a poset to be dual CR and hence partitionable. The flag h-vectors of strongly signable posets are therefore non-negative. It is proved that recursively shellable pose...

In this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. It is proved that an abelian group of finite exponent is strongly coatomic.

Let A =< A,≤A> and B =< B,≤B> be lattices such that A ∩ B is a filter in A and an ideal in B, and the orderings ≤A and ≤B coincide on A ∩ B. Then ≤A ∪ ≤B ∪ (≤A ◦ ≤B), is a lattice ordering on A ∪ B and the resulting lattice, called a sum of A and B, is denoted by A ⊕ B. The sum operation was introduced by Wroński [5], and its special case with A∩B = {1A} = {0B} by Troelstra [4]. In particular, ...

We show that universal algebras with factorable congruences such as rings with 1 and semirings with 0 and 1 enjoy some of the properties of universal algebras whose congruence lattices are distributive, such as the strict refinement property and a variant of Jónsson’s lemma. A universal algebra A is said to have factorable congruences if whenever A ∼= B × C and θ is a congruence on A, then θ = ...

Algebras on the natural numbers and their clones of term operations can be classified according to their descriptive complexity. We give an example of a closed algebra which has only unary operations and whose clone of term operations is not Borel. Moreover, we provide an example of a coatom in the clone lattice whose obvious definition via an ideal of subsets of natural numbers would suggest t...

A set A of nonnegative integers is recursively enumerable (r.e.) if A can be computably listed. It is shown that there is a rst order property, Q(X), de nable in E, the lattice of r.e. sets under inclusion, such that: (1) if A is any r.e. set satisfying Q(A) then A is nonrecursive and Turing incomplete; and (2) there exists an r.e. set A satisfying Q(A). This resolves a long open question stemm...