Left large subsets of free semigroups and groups that are not right large
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There are several notions of size for subsets of a semigroup \(S\) that originated in topological dynamics and are of interest because of their combinatorial applications as well as their relationship to the algebraic structure of the Stone–Čech compactification \(\beta S\) of \(S\). Among these notions are thick sets, central sets, piecewise syndetic sets, IP sets, and \(\Delta \) sets. Two related notions, namely \(C\) sets and \(J\) sets, arose in the study of combinatorial applications of the algebra of \(\beta S\). If the semigroup is noncommutative, then all of these notions have both left and right versions. In any semigroup, a left thick set must be a right \(J\) set (and of course a right thick set must be a left \(J\) set). We show here that for any free semigroup or free group on more than one generator, there is a set which satisfies all of the left versions of these notions and none of the right versions except \(J\). We also show that for the free semigroup on countably many generators, there is a left \(J\) set which is not a right \(J\) set.
KeywordsThick Central Piecewise syndetic \(C\)-set \(J\)-set Stone–Čech compactification Free semigroup Free group
N. Hindman acknowledges support received from the National Science Foundation (USA) via Grant DMS-1160566. L. L. Jones acknowledges support received from the Simons Foundation via Grant 210296.
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