# The word problem for some classes of Adian inverse semigroups

## TL;DR

This paper demonstrates that the finiteness of Schützenberger complexes for positive words in Adian inverse semigroups allows solving the word problem for specific classes of these semigroups, groups, and related structures.

## Contribution

It establishes a condition linking the finiteness of complexes of positive words to the overall finiteness, enabling solutions to the word problem in certain Adian inverse semigroups.

## Key findings

- Finiteness of complexes for positive words implies finiteness for all.
- Enables solving the word problem for specific Adian inverse semigroups.
- Applies to related Adian semigroups and groups.

## Abstract

We show that all of the Sch\"{u}tzenberger complexes of an Adian inverse semigroup are finite if the Sch\"{u}tzenberger complex of every positive word is finite. This enables us to solve the word problem for certain classes of Adian inverse semigroups (and hence for the corresponding Adian semigroups and Adian groups).