Preframe tensor product ($P\#Q$)

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.