Wreath Product Action on Generalized Boolean Algebras

Let $G$ be a finite group acting on the finite set $X$ such that the corresponding (complex) permutation representation is multiplicity free. There is a natural rank and order preserving action of the wreath product $G\sim S_n$ on the generalized Boolean algebra $B_X(n)$. We explicitly block diagonalize the commutant of this action.

Generalized Specker lattice-ordered groups and two types of distributivity

Let A be a lattice-ordered group, B a generalized Boolean algebra. The Boolean extension A B of A has been investigated in the literature; we will refer to A B as a generalized Specker lattice-ordered group (namely, if A is the linearly ordered group of all integers, then A B is a Specker lattice-ordered group). The paper establishes that some distributivity laws extend from A B to both A and B, and (under certain circumstances) also conversely.

The uncountable cofinality of the automorphism group of the countable universal distributive lattice

We show that the automorphism group of the countable universal distributive lattice has strong uncountable cofinality, and we adapt the method to deduce the strong uncountable cofinality of the automorphism group of the countable universal generalized boolean algebra.

On Birkhoff's Problem 73 For Monoids

Birkhoff in [2] poses the following problem:“Problem 73. Find necessary and sufficient conditions in order that the correspondence between the congruence relations and the (neutral) ideals of a lattice be one-one”.This problem has been solved by Areškin [l] and Hashimoto [3]. Essentially the conditions reduce to the r e quirement that the lattice be a generalized Boolean algebra.