Conservative groupoids recognize only regular languages

Abstract

The notion of recognition of a language by a finite semigroup can be generalized to recognition by finite groupoids, i.e. sets equipped with a binary operation '·' which is not necessarily associative. It is well known that L can be recognized by a groupoid iff L is context-free. But it is also known that some subclasses of groupoids can only recognize regular languages. For example, loops recognize exactly the regular open languages and Beaudry et al. described the largest class of groupoids known to recognize only regular languages. A groupoid H is said to be conservative if a·b is in {a, b} for all a, b in H. The main result of this paper is that conservative groupoids can only recognize regular languages. This class is incomparable with the one of Beaudry et al. so we are exhibiting a new sense in which a groupoid can be too weak to have context-free capabilities. © 2012 Springer-Verlag.