# Quasi-automatic semigroups

## TL;DR

This paper introduces quasi-automatic semigroups, a broad class with decidable word problems, rational presentations, and properties similar to automatic groups, expanding understanding of algebraic structures with computational benefits.

## Contribution

It defines quasi-automatic semigroups, shows their properties, and relates them to known classes like automatic groups, providing new insights into their computational and algebraic characteristics.

## Key findings

- Membership independent of generators
- Word problem decidable in exponential time
- Includes known semigroups and groups as special cases

## Abstract

A quasi-automatic semigroup is defined by a finite set of generators, a rational (regular) set of representatives, such that if a is a generator or neutral, then the graph of right multiplication by a on the set of representatives is a rational relation. This class of semigroups contains previously considered semigroups and groups (Sakarovitch, Epstein et al., Campbell et al.). Membership of a semigroup to this class does not depend on the choice of the generators. These semigroups are rationally presented. Representatives may be computed in exponential time. Their word problem is decidable in exponential time. They enjoy a property similar to the so-called Lipschitz property, or fellow traveler property. If graded, they are automatic. In the case of groups, they are finitely presented with an exponential isoperimetric inequality and they are characterized by the weak Lipschitz property.