Almost disjoint large subsets of semigroups |
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Authors: | Timothy J Carlson Jillian McLeod Dona Strauss |
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Institution: | a Department of Mathematics, Ohio State University, Columbus, OH 43210, USA b Department of Mathematics, Howard University, Washington, DC 20059, USA c Department of Mathematics, Mount Holyoke College, South Hadley, MA 01075, USA d Department of Pure Mathematics, University of Hull, Hull HU6 7RX, UK |
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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. |
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Keywords: | primary 54H15 secondary 03E05 22A15 05D10 |
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