Decision problems for inverse monoids presented by a single sparse relator

TL;DR

This paper investigates inverse monoids defined by a single sparse relator, proving the decidability of their word problem and characterizing related languages as deterministic context free or regular.

Contribution

It introduces the concept of sparse words in inverse monoids and establishes key language-theoretic properties and decidability results for these structures.

Findings

Word problem for sparse inverse monoids is decidable.

The identity language is deterministic context free.

The geodesic language is regular.

Abstract

We study a class of inverse monoids of the form M = Inv< X | w=1 >, where the single relator w has a combinatorial property that we call sparse. For a sparse word w, we prove that the word problem for M is decidable. We also show that the set of words in (X \cup X^{-1})^* that represent the identity in M is a deterministic context free language, and that the set of geodesics in the Schutzenberger graph of the identity of M is a regular language.