On local semirings induced by topologies: An algebraic approach to the Collatz conjecture
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.
Fields of Study & Subjects
Benford’s Law and Fraud Detection
Academic integrity and plagiarism
Probability and Statistical Research
Publication details
- Journal / source
- Applied General Topology
- Publisher
- Universitat Politecnica de Valencia
- Volume
- 27
- Issue
- 1
- Pages
- 24353–24353
- Publication date
- 18 February 2026
- Language
- EN