Is there a name for relations with this property?

Question

Is there a name for relations $\rho : X \rightarrow Y$ such that for all $x,x' \in X$ and all $y,y' \in Y$ we have that the following conditions $$xy \in \rho$$ $$x'y \in \rho$$ $$xy' \in \rho$$ imply that $$x'y' \in \rho?$$

Here's a couple of alternative characterizations.

1. Define $\rho(x) = \{y\in Y \,|\, xy \in \rho\}$ for all $x \in X$. Then for all relations $\rho : X \rightarrow Y,$ we have that $\rho$ has the property of interest iff for all $x,x' \in X$ it holds that either $\rho(x) = \rho(x')$ or $\rho(x) \cap \rho(x') = \emptyset$.

2. Call a relation $\kappa : X \rightarrow Y$ Cartesian iff there exist $A \subseteq X$ and $B \subseteq Y$ such that $\kappa = A \times B$. Call two relations $\kappa$ and $\kappa$' strongly disjoint iff their images are disjoint, and their "left-images" are also disjoint. Then for all relations $\rho : X \rightarrow Y,$ we have that $\rho$ has the property of interest iff it can be expressed as a strongly disjoint union of Cartesian relations $X \rightarrow Y$.

A few observations:

• If a relation has the property of interest, so too does its converse.
• Every function has the property (and thus, so too does its converse).
• If two relations have the property, their composition does, too; thus, we obtain a category.
• The property is preserved under arbitrary strongly disjoint unions.

Answer 1

Such relations are called rectangular in Section 5.2, page 669 of Andrei A. Bulatov, Víctor Dalmau, Towards a dichotomy theorem for the counting constraint satisfaction problem, Information and Computation, Volume 205, Issue 5, May 2007, Pages 651-678. They in fact define the property for higher arities, but it simplifies to your definition for binary relations.

Answer 2

If the relation is seen as an undirected bipartite graph with edges joining $X$ and $Y$, the axiom is that a path of length 3 closes to a cycle of length 4.

In this form it is easy to check that "are both related to the same element in the other set (or equal)" is an equivalence relation on $X$ and on $Y$, and that if $x$ is related to $y$, all elements of $X$ equivalent to $x$ are related to all elements of $Y$ equivalent to $y$. This is the condition for there to be quotients of the sets $X$ and $Y$ on which the relation descends to a one-to-one correspondence.

This can be expressed by saying that $\rho$ is/induces an "identification between quotients" of $X$ and $Y$. That property is not closed under composition as far as I can see, but the other properties are true.