La Catégorie des monoïdes résidués

Abstract

This work is our contribution to the study of the category of residuated ordered monoids. An ordered monoid is a poset (M;_) together with an associative operation _ and a neutral element e: A residuated ordered monoid is an ordered monoid (M;_; _; e) which is residuated in the sense that for all z; x 2 M; there is a biggest element y 2 M so that x _ y _ z and for all z; y 2 M; there is a biggest element x 2 M so that x _ y _ z: We have also studied the category of modules over residuated ordered monoids. We have shown that the category A-Mod of left A-modules and A-equivariant increasing maps have limits, colimits, some sublimits and subcolimits we have constructed free module and proved that there are enought injectives. We have shown that the category (A-Mod)_ of A-modules and residuated A- equivariant increasing maps for which right adjoints are A- equivariant have products indexed by ordered sets, and proved that there is equivalence between quotient congruences and quotient by quantics nucleus. The last result concerne the category of A-Alg; of A-algebras over residuated ordered monoid.We have shown that the forgetfull functor from A-Alg to A-Mod has left adjoint. Using Lambek’s calculus, we give applications of the theory of residuated ordered monoids in linguistic. We have proved in particular that residuated ordered monoids are models of computer lingustic. We have also mentioned our result concerning Rees congruences in semigroups which has been published.