Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Concerning three classes of non-Diophantine arithmetics   We present three classes of abstract prearithmetics, $\{\mathbf{A}_M\}_{M
\geq 1}$, $\{\mathbf{A}_{-M,M}\}_{M \geq 1}$, and $\{\mathbf{B}_M\}_{M > 0}$.
The first one is weakly projective with respect to the nonnegative real
Diophantine arithmetic
$\mathbf{R_+}=(\mathbb{R}_+,+,\times,\leq_{\mathbb{R}_+})$, the second one is
weakly projective with respect to the real Diophantine arithmetic
$\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$, while the third one is
projective with respect to the extended real Diophantine arithmetic
$\overline{\mathbf{R}}=(\overline{\mathbb{R}},+,\times,\leq_{\overline{\mathbb{R}}})$.
In addition, we have that every $\mathbf{A}_M$ and every $\mathbf{B}_M$ are a
complete totally ordered semiring, while every $\mathbf{A}_{-M,M}$ is not. We
show that the projection of any series of elements of $\mathbb{R}_+$ converges
in $\mathbf{A}_M$, for any $M \geq 1$, and that the projection of any
non-oscillating series series of elements of $\mathbb{R}$ converges in
$\mathbf{A}_{-M,M}$, for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M >
0$. We also prove that working in $\mathbf{A}_M$ and in $\mathbf{A}_{-M,M}$,
for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M>0$, allows to overcome a
version of the paradox of the heap.

18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys