Well quasi-orders arising from finite ordered semigroups

TL;DR

This paper investigates when homomorphisms from the semigroup of all words onto finite ordered semigroups induce well quasi-orders, establishing that this property is decidable and intrinsic to the semigroup itself.

Contribution

It proves the decidability of whether such homomorphisms induce well quasi-orders and shows this property depends solely on the ordered semigroup, not on the specific homomorphism.

Findings

Decidability of the property for finite ordered semigroups

The property is independent of the specific homomorphism

Characterization of when the induced order is a well quasi-order

Abstract

In 1985, Bucher, Ehrenfeucht and Haussler studied derivation relations associated with a given set of context-free rules. Their research motivated a question regarding homomorphisms from the semigroup of all words onto a finite ordered semigroup. The question is which of these homomorphisms induce a well quasi-order on the set of all words. We show that this problem is decidable and the answer does not depend on the homomorphism, but it is a property of the ordered semigroup.