Record number :

1581579

Title of article :

Almost disjoint large subsets of semigroups

Author/Authors :

Carlson، نويسنده , , Timothy J. and Hindman، نويسنده , , Neil and McLeod، نويسنده , , Jillian and Strauss، نويسنده , , Dona، نويسنده ,

Issue Information :

دوماهنامه با شماره پیاپی سال 2008

Pages :

12

From page :

433

To page :

444

Abstract :

There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick, central, syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in ( N , + ) any central set may be partitioned into infinitely many pairwise disjoint central sets. We extend this result to a large class of semigroups (including ( N , + ) ) by showing that if S is a semigroup in this class which has cardinality κ then any central set can be partitioned into κ many pairwise disjoint central sets. We also show that for this same class of semigroups, if there exists a collection of μ almost disjoint subsets of any member S, then any central subset of S contains a collection of μ almost disjoint central sets. The same statement applies if “central” is replaced by “thick”; and in the case that the semigroup is left cancellative, “central” may be replaced by “piecewise syndetic”. The situation with respect to syndetic sets is much more restrictive. For example, there does not exist an uncountable collection of almost disjoint syndetic subsets of N . We investigate the extent to which syndetic sets can be split into disjoint syndetic sets.

Keywords :

Almost disjoint , CENTRAL , Thick , Syndetic

Journal title :

Topology and its Applications

Serial Year :

2008

Link To Document :