A class of number theoretic finite semirings

Abstract: This work presents a novel structure of finite semirings constructed from factorization trees. We first establish the foundational elements of our theory by defining the largest and smallest prime divisors of a natural number. Utilizing these divisors, we construct factorization trees that inherently determine the lexicographic ordering of the factorizations. These trees enable us to define a projection function, which reinterprets factorization trees as decision trees for the projection of larger numbers. We introduce two recursive definitions for the projection function, $\pi_n(N)$, one based on the decision tree interpretation and another on divisors and lexicographic sorting. Tables of values for $\pi_i(N)$ provide insight into the behavior of these functions. Moreover, we identify the properties of $\pi_n(N)$, which are pivotal to the semiring operations defined on the set $[n]$. The addition and multiplication operations, denoted as $\oplus$ and $\otimes$ respectively, are defined through the iteration of the successor function. This leads to the establishment of $([n], \oplus, \otimes)$ as an abelian semiring. We also propose a notation to express congruence under projection and demonstrate the properties that validate the semiring definition. The abstract concludes with conjectures to simplify the computation of $\oplus$ and $\otimes$, supported by visual representations of addition and multiplication tables and successor graphs for various $n$ values. These conjectures, if proven, can significantly enhance computational efficiency and confirm the structure as an abelian semiring. Download-Link (pdf).

A note in inverse Galois theory. Abstract:
The note explores connections between inverse Galois theory and Hilbert irreducibility, presenting results in the form of theorems and lemmas. The main focus is on establishing conditions under which a finite group can be realized as a Galois group over the rational numbers. The note introduces a corresponding polynomial associated with a finite group and explores its irreducibility over specific varieties. The main results include Theorem 1, which establishes conditions for a group to be a Galois group, and Theorem 2, which demonstrates the equivalence between the applicability of Hilbert irreducibility to the corresponding polynomial and the realizability of every finite group as a Galois group over the rational numbers. The note concludes with corollaries and lemmas supporting the main theorems. It can be downloaded as pdf from here.