The Inverse Semigroup Theory of Elementary Arithmetic

TL;DR

This paper explores the inverse semigroup structures arising from elementary arithmetic operations on natural numbers, introducing a novel inverse monoid that connects number theory with semigroup theory.

Contribution

It introduces the arithmetic inverse monoid A, generalizing classic inverse monoids, and links semigroup concepts with number-theoretic structures like the Chinese remainder theorem.

Findings

Introduction of the arithmetic inverse monoid A

Connection between normal forms and number-theoretic concepts

Identification of prime-order polycyclic monoids as generators

Abstract

We curry the elementary arithmetic operations of addition and multiplication to give monotone injections on N, and describe & study the inverse monoids that arise from also considering their generalised inverses. This leads to well-known classic inverse monoids, as well as a novel inverse monoid (the 'arithmetic inverse monoid' A) that generalises these in a natural number-theoretic manner. Based on this, we interpret classic inverse semigroup theoretic concepts arithmetically, and vice versa. Composition and normal forms within A are based on the Chinese remainder theorem, and a minimal generating set corresponds to all prime-order polycyclic monoids. This then gives a close connection between Nivat & Perot's normal forms for polycyclic monoids, mixed-radix counting systems, and p-adic norms & distances.