In this paper, we define and study a class of monoids ordered by a two sided version of Nambooripad order. These monoids, quasi-inverse in some sense, are equipped with a variation of McAlister and Reilly premorphisms. The resulting category is shown to be a super-category of the category of monoids equipped with morphisms. An expansion construction, à la Birget and Rhodes, that lifts every monoid morphism to a quasi-inverse monoid premorphism, is provided. This shows that, in some sense, the classical notion of language recog- nizability by monoid can be generalized to a richer notion of language quasi-recognizability by quasi-inverse monoid. While former studies of inverse monoids in the context of language theory rather led to negative results as the class of definable languages by means of inverse monoids collapses, the new framework proposed here reopens the way towards the integration of the mathematical rich- ness of inverse monoid theory in algebraic studies of formal language theory. Nota : this research report is unpublished and will probably remain as such. Ongoing developments, within language theory, of the notion of quasi- recognizability tell us that definitions proposed in this report are yet not as potentially stable as one could expect.
