  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.
