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Concerning three classes of non-Diophantine arithmetics

### Michele Caprio, Andrea Aveni and Sayan Mukherjee

Vol. 15 (2022), No. 5, 763–774
##### Abstract

We present three classes of abstract prearithmetics, ${\left\{{A}_{M}\right\}}_{M\ge 1}$, ${\left\{{A}_{-M,M}\right\}}_{M\ge 1}$, and ${\left\{{B}_{M}\right\}}_{M>0}$. The first is weakly projective with respect to the nonnegative real Diophantine arithmetic ${R}_{+}=\left({ℝ}_{+},+,×,{\le }_{{ℝ}_{+}}\right)$, the second is weakly projective with respect to the real Diophantine arithmetic $R=\left(ℝ,+,×,{\le }_{ℝ}\right)$, while the third is exactly projective with respect to the extended real Diophantine arithmetic $\overline{R}=\left(\overline{ℝ},+,×,{\le }_{\overline{ℝ}}\right)$. In addition, we have that every ${A}_{M}$ and every ${B}_{M}$ is a complete totally ordered semiring, while every ${A}_{-M,M}$ is not. We show that the projection of any series of elements of ${ℝ}_{+}$ converges in ${A}_{M}$, for any $M\ge 1$, and that the projection of any nonindeterminate series of elements of $ℝ$ converges in ${A}_{-M,M}$, for any $M\ge 1$, and in ${B}_{M}$, for all $M>0$. We also prove that working in ${A}_{M}$ and in ${A}_{-M,M}$, for any $M\ge 1$, and in ${B}_{M}$, for all $M>0$, allows us to overcome a version of the paradox of the heap.

##### Keywords
non-Diophantine arithmetics, convergence of series, paradox of the heap
Primary: 03H15
Secondary: 03C62
##### Milestones
Received: 6 January 2021
Revised: 27 October 2021
Accepted: 11 March 2022
Published: 3 March 2023

Communicated by Kenneth S. Berenhaut
##### Authors
 Michele Caprio Department of Statistical Science Duke University Durham, NC United States Andrea Aveni Department of Statistical Science Duke University Durham, NC United States Sayan Mukherjee Department of Statistical Science Duke University Durham, NC United States