Name of a relation R⊆A×B s.t. ∀a'≤a, b≤b': aRb ⇒ a'Rb'

Question

Given two partially ordered sets $A$ and $B$, is there a name for a binary relation $R⊆A×B$ satisfying this property:

Whenever $(a,b)∈R$, then also

• $\forall a' \le a, \forall b' \ge b: (a',b')\in R$

Answer 1

In the context of enriched category theory, such relation is called module, distributor or profunctor (less common). You can check the motivation in the book below that discusses the particular case or posets.

http://www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf