Eraser morphisms and membership problem in groups and monoids

TL;DR

This paper introduces eraser morphisms and the concept of fragile words, characterizing their images in free groups and inverse monoids, and establishing decidability of membership problems and algorithmic properties for certain classes of monoids.

Contribution

It extends the theory of fragile words via eraser morphisms to inverse monoids and proves decidability results for membership problems in these algebraic structures.

Findings

The image of the eraser morphism in free groups is characterized.

Membership problem for the eraser morphism's image in free inverse monoids is decidable.

Finite-${\

Abstract

We develop the theory of fragile words by introducing the concept of eraser morphism and extending the concept to more general contexts such as (free) inverse monoids. We characterize the image of the eraser morphism in the free group case, and show that it has decidable membership problem. We establish several algorithmic properties of the class of finite-${\cal{J}}$-above (inverse) monoids. We prove that the image of the eraser morphism in the free inverse monoid case (and more generally, in the finite-${\cal{J}}$-above case) has decidable membership problem, and relate its kernel to the free group fragile words.