Work

On local semirings induced by topologies: An algebraic approach to the Collatz conjecture

Year: 2026

Type: article

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.

Source: Applied General Topology

DOI: 10.4995/agt.24353

Authors: Angel Guale, Jorge Vielma

Institution: Escuela Superior Politecnica del Litoral

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Escuela Superior Politécnica del Litoral

Award: G1669897504