Tensor products for bounded posets revisited

The category BPC of bounded posets and so-called cut continuous maps has concrete products, and the Dedekind-MacNeille completion gives rise to a reflector from BPC to the full subcategory CLJ of complete lattices and join-preserving maps. Like CLJ, the category BPC has a functional internal hom-functor in the sense of Banaschewski and Nelson. But, in contrast to CLJ, arbitrary universal bimorphisms do not exist in BPC. However, a natural tensor product is defined in terms of so-called G-ideals, such that the desired universal property holds at least for BPC-morphisms into complete lattices. Moreover, this tensor product is associative and distributes over (cartesian) products. The tensor product of an arbitrary family of bounded posets is isomorphic to that of their normal completions; hence, restricted to the subcategory CLJ, it agrees with the usual one.