Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Problems and results on intersections of product sets and sumsets in semigroups

Melvyn B. Nathanson

Abstract

For every subset $A$ of a semigroup $S$, let $A^h$ be the set of all products of $h$ elements of $S$. If $(A)_{q\in Q}$ is a family of subsets of $S$, then $A = \bigcap_{q \in Q} A_q$ satisfies $A^h \subseteq \bigcap_{q \in Q} A_q^h$. The product intersection set $H(A_q) = \left\{h \in \mathbf{N}: A^h = \bigcap_{q \in Q} A_q^h \right\}$ is investigated.

Problems and results on intersections of product sets and sumsets in semigroups

Abstract

For every subset of a semigroup , let be the set of all products of elements of . If is a family of subsets of , then satisfies . The product intersection set is investigated.

Paper Structure

This paper contains 6 sections, 22 theorems, 176 equations.

Table of Contents

  1. The product and sum intersection sets
  2. Intersections of sequences of sets
  3. Representation functions and intersection sets
  4. Product intersection sets in groups
  5. Intersections of product intersection sets
  6. $R$-module transformations

Key Result

Theorem 1

For every positive integer $h_0$, there exists $(A_q)_{q=1}^{\infty} \in \mathcal{F}_{ \mathbf N }^*(\mathbf Z)$ such that For every integer $d \geq 2$, there exists $(A_q)_{q=1}^{\infty} \in \mathcal{F}_{ \mathbf N }^*(\mathbf Z)$ such that $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (43)

  • Theorem 1: Marques and Nathanson marq-nath26
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 33 more