On lax protomodularity for Ord-enriched categories

Our main focus concerns a possible lax version of the algebraic property of protomodularity for Ord-enriched categories. Having in mind the role of comma objects in the enriched context, we consider some of the characteristic properties of protomodularity with respect to comma objects instead of pullbacks. We show that the equivalence between protomodularity and certain properties on pullbacks also holds when replacing conveniently pullbacks by comma objects in any finitely complete category enriched in Ord, and propose to call lax protomodular such Ord-enriched categories. We conclude by studying this sort of lax protomodularity for the category OrdAb of preordered abelian groups, equipped with a suitable Ord-enrichment, and show that OrdAb fulfills the equivalent lax protomodular properties with respect to the weaker notion of precomma object; we call such categories lax preprotomodular.