Dioid partitions of groups

Partitions and Permutation Groups

We show that non-trivial extremely amenable topological groups are essentially the same thing as permutation models of the Boolean prime ideal theorem that do not satisfy the axiom of choice. Both are described in terms of partition properties of group actions.

On Zero-sum Partitions of Abelian Groups

In this paper, confirming a conjecture of Kaplan et al., we prove that every abelian group G, which is of odd order or contains exactly three involutions, has the zerosum-partition property. As a corollary, every tree with |G| vertices and at most one vertex of degree 2 is G-anti-magic.

Non-crossing partitions for classical reflection groups

We introduce analogues of the lattice of non-crossing set partitions for the classical reeection groups of type B and D. The type B analogues ((rst considered by Montenegro in a diierent guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved. In both cases, they are EL-labellable ranked lattices with symmetric chain dec...

Non-crossing partitions for arbitrary Coxeter groups

k-Efficient partitions of graphs

A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this ...