# Green's Relations in Finite Transformation Semigroups

## TL;DR

This paper investigates the complexity of Green's relations in finite transformation semigroups, establishing an exponential lower bound for the maximal chain length of equivalence classes, with implications for automata theory.

## Contribution

It provides the first exponential lower bound for the maximal chain length of Green's relations in finite transformation semigroups with a fixed alphabet.

## Key findings

- Exponential lower bound for chain length of Green's relations.
- Construction applicable to any generator set, including constant alphabet.
- Results extend to automata and their syntactic semigroups.

## Abstract

We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.