Binary equality sets are generated by two words

TL;DR

This paper proves that the equality language of two non-periodic binary morphisms can be generated by at most two words, leading to a simple test set for any binary language.

Contribution

It establishes that the equality language of two non-periodic binary morphisms is generated by at most two words, with specific properties when the rank is two.

Findings

Equality language generated by two words for non-periodic binary morphisms

Generators start and end with different letters when rank is two

Any binary language has a test set of at most two words

Abstract

We show that the equality language of two non-periodic binary morphisms is generated by at most two words. If its rank is two, then the generators start (and end) with different letters. This in particular implies that any binary language has a test set of cardinality at most two.