Monoids are equivalent to automata
A language is called -recognisable if there is some homomorphism of -monoids
such that has finite universe, and which recognizes in the sense that membership is uniquely determined by . The following theorem shows that in the special (but not all that special) case of languages that contain only -words, this notion coincides with the notion of -regularity.
Theorem. A language is -recognisable if and only if it is -regular.
Proof. For define to be the finite nonempty words that are mapped by to , and define to be the -words that are mapped by to . It is easy to see that each language is a regular language of finite words, because it is recognised by a finite monoid, namely the monoid obtained from by ignoring products for infinite words. Furthermore, using the Ramsey theorem in the same way as in the previous lemma, we conclude that is equal to the -regular language
with the union ranging over elements which satisfy . Every set of -words recognised by is a finite union of languages of the form , and therefore the left-to-right implication follows by closure of -regular languages under finite union.
For the right-to-left implication, suppose that is recongised by a nondeterministic Büchi automaton with states . Define
defined as follows. The empty word gets mapped to . Infinite words are mapped to , namely an infinite word is mapped to the set of those states such that the word admits an accepting run that begins with . Finite words are mapped to in the same way as in the proof of complementation for Büchi automata. It is not difficult to see that this mapping is compositional, and therefore its image can be equipped with the structure of an -monoid which makes it into a homomorphism.