Po-groups and hypergroups in a topos

Abstract

This paper deals with two constructions in topos theory: po-groups and hypergroups. After a deep analysisof these, we restrict our attention to find a hypergroup associated to a po-group G in a topos E . The method thatwe use here is based on the Mitchell–B´enabou language. Then, we show that on the negative and positive cones of apo-group G in E, the left and right translations are hyperhomomorphisms in E. Our aim is to find two faithful and leftexact functors from the category of po-groups in E to the (smallest in some sense) finitely complete category containinghypergroups in E . A version of this result is also presented on the category of lattices in E instead of po-groups. Thisversion recovers filters and ideals of a lattice in E by means of hyperoperations. We will finish the manuscript bytransforming the Heyting algebra structure of the subobject classifier Ω of E to a hypergroup in E .