Preframe

A preframe is a partially ordered set with finite meets and nonempty directed joins, with finite meets distributing over directed joins. Its homomorphisms preserve these operations. Forgetting the arbitrary joins of a frame leaves a preframe.

The tensor represents maps $P\times Q\to R$ preserving finite meets and directed joins in each variable separately. Its pure generators $p\#q$ satisfy the corresponding relations in each variable. The unit is the free preframe on one generator, the bottom of the frame of truth values. For frames, this tensor is their coproduct, with $p\#q=i(p)\vee j(q)$; it must not be confused with the rectangle $i(p)\wedge j(q)$.

For a commutative monoid object $(M,\ast ,e)$ in preframes, the indicated subset is a frame. Its bottom is $e$, its binary join is $\ast$, and its directed joins and finite meets are inherited. Every homomorphism from a frame regarded as a join monoid factors uniquely through this subset. This gives a coreflective subcategory.