Regular Nests

A language over a finite alphabet X is regular if it can be accepted by a finite automaton. It is known that the family of all regular languages L3 over X is closed under catenation. This paper is a study of the regular property of the catenation of two languages. We call a family of languages L a regular nest if the catenation of any two languages in L is regular. We construct some natural regular nests which contain non-regular languages. We also investigate some languages which catenate with any non-empty languages on the right (left) can never be regular. Such languages we call them absolutely right (left) non-regular languages. We give an example of absolutely right non-regular language which is not an absolutely left non-regular. We show that for X = {a, b, c}, the well known context-sensitive language {anbncn|n ≥ 1}, and the case when |X| = 2, Dyck language D and the balanced language H are absolutely right non-regular languages.