Is there a name for relations with this property, and the category of them? The following math.stackexchange question asked whether there is a name for a certain sort of relation.  I have become interested in the question, and no one suggested a name there, so I am asking here instead.  Furthermore I would be interested in the name of the category of such relations.
https://math.stackexchange.com/questions/391108/is-there-a-name-for-relations-with-this-property
I repeat the specification from the above link.  The relations are those of the form $\rho : X \rightarrow Y$ 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 $$x'y' \in \rho$$
By "category of such relations" I mean the category whose objects are such relations $\rho : X_\rho \rightarrow Y_\rho$, and whose morphisms are pairs of functions $(f_X : X_\rho \to X_{\rho'}, \,\, f_Y : Y_\rho \to Y_{\rho'})$ that preserve relatedness, i.e. $xy \in \rho \Rightarrow f_X(x)f_Y(y) \in \rho'$.
I am considering these concepts in the context of John Reynold's work on relational parametricity in type theory.
 A: The set of all relations $\alpha\subset X\times Y$ is a semiheap with respect to the ternary operation $[\alpha_1,\alpha_2,\alpha_3]=\alpha_1\alpha_2^{-1}\alpha_3$ ($\alpha^{-1}$ is $\alpha^{*}$in notation of Tom Ellis and $\alpha^{op}$ by Todd Trimble). Your relation yields $[\rho,\rho,\rho]=\rho$ and is called an idempotent (http://en.wikipedia.org/wiki/Semiheap).
A: I have come across such relations in the past, and (in my head) have referred to them as jigsaw relations. The idea is probably best conveyed in the form of a picture, but I will try to explain my thinking.
Let $\rho\colon X\to Y$ be a relation arising from jigsaw pieces in the following way. Think of the elements of $X$ as jigsaw pieces with a 'tab' sticking out of the right edge, the elements of $Y$ as jigsaw pieces with a 'hole' in the left edge, and $xy\in\rho$ as meaning "the tab of $x$ fits exactly into the hole of $y$".
Claim: such a relation satisfies $xy,x'y,xy'\in\rho\implies x'y'\in\rho$.
Proof: if $xy\in\rho$ and $x'y\in\rho$ then the tab of $x'$ must be identical to the tab of $x$ (because they both fit exactly into the hole of $y$), so if $xy'\in\rho$ as well then the tab of $x'$ fits exactly into the hole of $y'$ because the tab of $x$ does. That is, $x'y'\in\rho$ follows
from the other three conditions.
A: These have probably been invented many times, but they are most typically called difunctional relations.  They were introduced and named that by Jacques Riguet in 1948, Relations binaires, fermetures, correspondances de Galois.
A nice paper on them is Lambek's Goursat's Theorem and the Zassenhaus Lemma, which shows that when you have a Mal'cev term (which includes groups, rings, Lie algebras, etc.) that every relation is difunctional, and you can use that to give a unified proof of the Jordan-Hölder theorem.  (The unified proof is not new with that paper, but it shows how to effectively use difunctionality.)
