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arXiv · 2206.07412

The Inverse Semigroup Theory of Elementary Arithmetic

Peter M. Hines
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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.

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Peter M. Hines. 2022-06-15. The Inverse Semigroup Theory of Elementary Arithmetic. https://arxiv.org/abs/2206.07412

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