Left large subsets of free semigroups and groups that are not right large

Abstract

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–Cech 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.