%0 Journal Article %1 proceedings2022081033 %A Caprio, Michele %A Aveni, Andrea %A Mukherjee, Sayan %D 2022 %J Proceedings %K Arithmetics Non-Diophantine %N 1 %R 10.3390/proceedings2022081033 %T Concerning Two Classes of Non-Diophantine Arithmetics %U https://www.mdpi.com/2504-3900/81/1/33 %V 81 %X We present two classes of abstract prearithmetics, AMM≥1 and BMM>0. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic R+=(R+,+,×,≤R+), and the second one is projective with respect to the extended real Diophantine arithmetic R¯=(R¯,+,×,≤R¯). In addition, we have that every AM and every BM is a complete totally ordered semiring. We show that the projection of any series of elements of R+ converges in AM, for any M≥1, and that the projection of any non-indeterminate series of elements of R converges in BM, for all M>0. We also prove that working in AM, for any M≥1, and in BM, for all M>0, allows to overcome a version of the paradox of the heap.

@article{proceedings2022081033, abstract = {We present two classes of abstract prearithmetics, {AM}M≥1 and {BM}M>0. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic R+=(R+,+,×,≤R+), and the second one is projective with respect to the extended real Diophantine arithmetic R¯=(R¯,+,×,≤R¯). In addition, we have that every AM and every BM is a complete totally ordered semiring. We show that the projection of any series of elements of R+ converges in AM, for any M≥1, and that the projection of any non-indeterminate series of elements of R converges in BM, for all M>0. We also prove that working in AM, for any M≥1, and in BM, for all M>0, allows to overcome a version of the paradox of the heap.}, added-at = {2024-11-12T13:17:42.000+0100}, article-number = {33}, author = {Caprio, Michele and Aveni, Andrea and Mukherjee, Sayan}, biburl = {https://puma.scadsai.uni-leipzig.de/bibtex/24f3652e1c2a1d84aa07d7cde33a4cdf2/scadsfct}, doi = {10.3390/proceedings2022081033}, interhash = {1f8dc80702f78e624a1e331c506b77b3}, intrahash = {4f3652e1c2a1d84aa07d7cde33a4cdf2}, issn = {2504-3900}, journal = {Proceedings}, keywords = {Arithmetics Non-Diophantine}, number = 1, timestamp = {2024-11-12T13:17:42.000+0100}, title = {Concerning Two Classes of Non-Diophantine Arithmetics}, url = {https://www.mdpi.com/2504-3900/81/1/33}, volume = 81, year = 2022 }