Cartesian product of combinatorially rich sets- algebraic, elementary and dynamical approaches

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-11-23 DOI:10.1016/j.topol.2024.109148

Pintu Debnath

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引用次数: 0

Abstract

Using the methods of topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. In [Fund. Math 199 (2008)], D. De, N. Hindman, and D. Strauss introduced the notion of a C-set, satisfying the strong central sets theorem. In [Topology Proc. 35 (2010)], using the algebraic structure of the Stone-Čech compactification of a discrete semigroup, N. Hindman and D. Strauss proved that the Cartesian product of two C-sets is a C-set. S. Goswami has proved the same result using the elementary characterization of C-sets. In this article, we will prove that the product of two C-sets is a C-set, using the dynamical characterization of C-sets. Recently, S. Goswami has proved that the Cartesian product of two CR-sets is a CR-set, which was a question posed by N. Hindman, H. Hosseini, D. Strauss, and M. Tootkaboni in [Semigroup Forum 107 (2023)]. Here we also prove that the Cartesian product of two essential CR-sets is an essential CR-set.

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.