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A combinatorial approach to the stronger Central Sets Theorem for semigroups

Pintu Debnath

TL;DR

This work provides a new combinatorial proof of the stronger Central Sets Theorem ($SCST$) for discrete semigroups. It first develops a commutative version by constructing a hierarchical $\alpha$–$H$ scheme on finite families of sequences and leveraging that central sets induce $J$-sets, together with the $Hales$–$Jewett$ theorem to embed combinatorial lines into central sets. It then extends to noncommutative semigroups by proving that piecewise syndetic sets are $J$-sets via a noncommutative $Hales$–$Jewett$ argument and by inductively assembling parameters $m(F)$, $\alpha(F)$, and $\tau(F)$ to ensure the products $\prod_{i=1}^n x(m(G_i),\alpha(G_i),\tau(G_i),f_i)$ belong to $A$ for all chains of finite families. Overall, the paper provides a purely combinatorial derivation of SCST in both commutative and noncommutative settings, highlighting the pivotal roles of $J$-sets and the $Hales$–$Jewett$ theorem in semigroup Ramsey theory.

Abstract

H. Furstenberg introduced the notion of central sets in terms of topological dynamics and established the famous Central Sets Theorem. Later in [A new and stronger Central Sets Theorem, Fund. Math. 199 (2008), 155-175], D. De, N. Hindman, and D. Strauss established a stronger version of the Central Sets Theorem that uses the algebra of the Stone-\v Cech compactification of discrete semigroups. In this article, We will provide a new and combinatorial proof of the stronger Central Sets Theorem.

A combinatorial approach to the stronger Central Sets Theorem for semigroups

TL;DR

This work provides a new combinatorial proof of the stronger Central Sets Theorem () for discrete semigroups. It first develops a commutative version by constructing a hierarchical – scheme on finite families of sequences and leveraging that central sets induce -sets, together with the – theorem to embed combinatorial lines into central sets. It then extends to noncommutative semigroups by proving that piecewise syndetic sets are -sets via a noncommutative – argument and by inductively assembling parameters , , and to ensure the products belong to for all chains of finite families. Overall, the paper provides a purely combinatorial derivation of SCST in both commutative and noncommutative settings, highlighting the pivotal roles of -sets and the – theorem in semigroup Ramsey theory.

Abstract

H. Furstenberg introduced the notion of central sets in terms of topological dynamics and established the famous Central Sets Theorem. Later in [A new and stronger Central Sets Theorem, Fund. Math. 199 (2008), 155-175], D. De, N. Hindman, and D. Strauss established a stronger version of the Central Sets Theorem that uses the algebra of the Stone-\v Cech compactification of discrete semigroups. In this article, We will provide a new and combinatorial proof of the stronger Central Sets Theorem.
Paper Structure (3 sections, 10 theorems, 55 equations)

This paper contains 3 sections, 10 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Commutative stronger central set theorem
  3. Noncommutative stronger Central set theorem

Key Result

Theorem 1.2

Let $A$ be a central subset of $\mathbb{N}$, let $k\in\mathbb{N}$, and for each $i\in\{1,2,\dots,k\}$ let $\langle y_{i,n}\rangle_{n=1}^{\infty}$ be a sequence in $\mathbb{Z}$. Then there exist sequences $\langle a_n\rangle_{n=1}^{\infty}$ in $\mathbb{N}$ and $\langle H_n\rangle_{n=1}^{\infty}$ in $

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4: Hales--Jewett
  • Lemma 2.5
  • ...and 8 more