The Rank and Hanna Neumann Property of Some Submonoids of a Free Monoid

TL;DR

This paper investigates the rank of intersections of finitely generated submonoids of free monoids, providing new results for automata accepted submonoids and conditions for the Hanna Neumann property.

Contribution

It extends previous work by determining the rank of certain submonoids accepted by semi-flower automata and establishing conditions for the Hanna Neumann property in automaton products.

Findings

Rank of submonoids accepted by semi-flower automata with two bpi's determined.

Sufficient conditions for the Hanna Neumann property in automaton products identified.

New insights into the structure of submonoids in free monoids obtained.

Abstract

This work aims at further investigations on the work of Giambruno and Restivo to find the rank of the intersection of two finitely generated submonoids of a free monoid. In this connection, we obtain the rank of a finitely generated submonoid of a free monoid that is accepted by semi-flower automaton with two bpi's. Further, when the product automaton of two deterministic semi-flower automata with a unique bpi is semi-flower with two bpi's, we obtain a sufficient condition on the product automaton in order to satisfy the Hanna Neumann property.