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

• Guale, Angel ^[1] ; Vielma, Jorge ^[1]
  1. [1] Escuela Superior Politecnica del Litoral

     Escuela Superior Politecnica del Litoral

     Guayaquil, Ecuador

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 27, Nº. 1, 2026
• Idioma: inglés
• DOI: 10.4995/agt.24353
• Enlaces
  • Texto completo
• Resumen

  • 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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