Polymultisets, Multisuccessors, and Multidimensional Peano Arithmetics

TL;DR

This paper introduces a new concept of natural multidimensional numbers using polymultisets and constructs a generalized Peano arithmetic, establishing a foundation for multidimensional number systems.

Contribution

It defines polymultisets and multisuccessors, and formulates axioms for natural multidimensional numbers, creating a new algebraic framework.

Findings

Defined polymultisets as m-ary multirelations

Established multisuccessor and multipredecessor functions

Constructed a commutative semiring of 2-numbers

Abstract

The goal of this paper is introduction of a concept of natural multidimensional numbers and to construct a generalized Peano arithmetic of these multidimensional numbers. For this purpose we define a polymultiset as a special set-like form of m-ary multirelation. In addition, multisuccessor and multipredecessor functions on polymultisets are determined. By introducing Peano-like axioms for natural multidimensional numbers, we build a commutative semiring of 2-numbers <P2, +, \cdot, 02, 100>.