A Combinatorial Property of Ideals in Free Profinite Monoids
Benjamin Steinberg
TL;DR
This paper demonstrates that in free profinite monoids, every regular element generates a prime ideal, providing an elementary proof for a key property previously established using symbolic dynamics.
Contribution
It offers a new, elementary proof that all regular elements generate prime ideals in free profinite monoids, including the minimal ideal.
Findings
Every regular element generates a prime ideal.
The minimal ideal is prime.
Elementary proof replaces symbolic dynamics techniques.
Abstract
We prove every regular element of a free profinite monoid generates a prime ideal; in particular the minimal ideal is prime. The latter result was first proved by Almeida and Volkov using techniques from symbolic dynamics; our proof is elementary.
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Taxonomy
Topicssemigroups and automata theory · Computability, Logic, AI Algorithms · Advanced Topology and Set Theory