Po-groups and hypergroups in a topos

Authors: ALI MADANSHEKAF, ZEINAB KANJANZADEH

Abstract: This paper deals with two constructions in topos theory: po-groups and hypergroups. After a deep analysis of these, we restrict our attention to find a hypergroup associated to a po-group $G$ in a topos $\mathcal{E}$. The method that we use here is based on the Mitchell--B$\acute{{\rm e}}$nabou language. Then, we show that on the negative and positive cones of a po-group $G$ in $\mathcal{E},$ the left and right translations are hyperhomomorphisms in $\mathcal{E}.$ Our aim is to find two faithful and left exact functors from the category of po-groups in $\mathcal{E}$ to the (smallest in some sense) finitely complete category containing hypergroups in $\mathcal{E}$. A version of this result is also presented on the category of lattices in $\mathcal{E}$ instead of po-groups. This version recovers filters and ideals of a lattice in $\mathcal{E}$ by means of hyperoperations. We will finish the manuscript by transforming the Heyting algebra structure of the subobject classifier $\Omega$ of $\mathcal{E}$ to a hypergroup in $\mathcal{E}$.

Keywords: Po-group, hypergroup, topos, intuitionistic logic

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