Zeroth-rank operation and non transitive numbers. Nulranga operacio kaj netransitivaj nombroj. Operazione di rango zero e numeri non transitivi

TL;DR

This paper introduces a new operation called incrementation, explores its properties, defines a new set of numbers called Escherian numbers where decrementation is closed, and investigates their non-transitive order relations.

Contribution

It defines incrementation consistent with hyper-operations, introduces Escherian numbers with non-transitive order, and extends incrementation to complex numbers, expanding the understanding of non-transitive number systems.

Findings

Incrementation is consistent with Ackermann's function.

Escherian numbers are not transitive under their pseudoorder.

Decrementation is closed on the set of Escherian numbers and extended to complex numbers.

Abstract

Observing the existing relationships between the elementary operations of addition, multiplication (iteration of additions) and exponentiation (iteration of multiplications), a new operation (named incrementation) is defined, consistently with these laws and such that addition turns out to be an iteration of incrementations. Incrementation turns out to be consistent with Ackermann's function. After defining the inverse operation of incrementation (named decrementation), we observe that R is not closed under it. So a new set of numbers is defined (named E, Escherian numbers), such that decrementation is closed on it. After defining the concept of pseudoorder (analogous to the order, but not transitive), it is shown that Escherian numbers are not transitive. Then addition and multiplication on E are analysed, and a correspondence between E and C is found. Finally, incrementation is extended to C, in such a way that decrementation is closed on C too. English keywords: hyper-operations, incrementation, zeration, Ackermann function, intransitive order, not transitive order, intransitive numbers, non transitive numbers, not transitive numbers, new number sets.