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\(D\)-sets in arbitrary semigroup. (English) Zbl 1508.05167

Summary: We define the notion of \(D\)-set in an arbitrary semigroup, and with some mild restrictions we establish its dynamical and combinatorial characterizations. Assuming a weak form of cancellation in semigroups we have shown that the Cartesian product of finitely many \(D\)-sets is a \(D\)-set. A similar partial result has been proved for Cartesian product of infinitely many \(D\)-sets. Finally, in a commutative semigroup we deduce that \(D\)-sets (with respect to a Følner net) are \(C\)-sets.

MSC:

05D10 Ramsey theory
20M99 Semigroups
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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