On local semirings induced by topologies: An algebraic approach to the Collatz conjecture
Submitted: 2025-07-21
|Accepted: 2025-11-10
|Published: 2026-02-18
Copyright (c) 2025 Angel Guale, Jorge Vielma
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
local semiring, additively idempotent semiring, Collatz, primal topology, functional Alexandroff space
Supporting agencies:
This research has been funded by ESPOL - Escuela Superior Politécnica del Litoral through project number FCNM-006-2024.
Abstract:
We present an algebraic approach to the Collatz conjecture by studying the topology τf on ℕ induced by the Collatz function f, where the open sets θ ⊂ ℕ satisfy f-1 ( θ ) ⊂ θ . This topology, known as \emph{primal topology}, turns τf into a commutative semiring. We prove that the Collatz conjecture holds if and only if τf is local. More generally, we show that any compact primal topology corresponds to a semiring that decomposes as a finite direct sum of certain local semirings and that primal compactness connectedness characterises locality. In addition, we establish that a topological space is not w-R0 if and only if its associated semiring of open sets has a unique maximal ideal such that it is an avoidance ideal of a closed set.
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