In the context of the $\mathcal{ OCA}$ associated to an ${ \mathcal{ AKS}}$ we introduce a closure operator and two associated maps that replace the closure and the maps defined in \cite{ kn:ocar}. We were motivated by the search of a full adjunction to the original implication map. We show that all the constructions from $\mathcal{ OCA}$s to triposes developped in \cite{ kn:ocar} can be also implemented in the new situation.
Categories: math.LO