The Intersection of $3$-Maximal Submonids

TL;DR

This paper explores the structure of the intersection of two 3-maximal monoids, providing a characterization for the case when both monoids are 3-maximal, a topic with limited prior understanding.

Contribution

It offers the first detailed characterization of the intersection of two 3-maximal monoids, advancing knowledge in the structure of such algebraic objects.

Findings

Characterization of the intersection for two 3-maximal monoids

Insights into the structure of 3-maximal monoids

Foundation for further research on monoid intersections

Abstract

Very little is known about the structure of the intersection of two $k$-generated monoids of words, even for $k=3$. Here we investigate the case of $k$-maximal monoids, that is, monoids whose basis of cardinality $k$ cannot be non-trivially decomposed into at most $k$ words. We characterize the intersection in the case of two $3$-maximal monoids.