Is there a name for those relations that behave a bit like $<$?

Question

Consider a fixed but arbitrary preordered set $X$. Is there a name for those binary relations $R$ on $X$ satisfying the following? They seem to show up a lot. (Note that every such $R$ is necessarily transitive.)

1. $R(x,y) \rightarrow x \lesssim y$
2. $R(x,y) \wedge (y \lesssim z) \rightarrow R(x,z)$
3. $(x \lesssim y) \wedge R(y,z) \rightarrow R(x,z)$

Examples.

• The strict $(<)$ and non-strict $(\lesssim)$ order relations on any preordered set $X$.

• Let $I$ denote an arbitrary non-empty set. Then the set $\mathbb{R}^I$ of functions $I \rightarrow \mathbb{R}$ is partially ordered in the obvious way via the pointwise order: $$f \leq g \iff (\forall i \in I)(f(i) \leq g(i)).$$ Now define a binary relation $R$ on $\mathbb{R}^I$ as follows. $$R(f,g) \iff (\forall i \in I)(f(i) < g(i)).$$ Then $R$ satisfies the conditions of interest. More generally, we can replace $\mathbb{R}$ with any preordered set $X$, and we can replace $<$ with any binary relation on $X$ satisfying 1,2 and 3.

• (This is the example I'm most interested in.) Let $X$ denote a metric space. Then its powerset $\mathcal{P}(X)$ is partially ordered by inclusion. Define $R$ on $\mathcal{P}(X)$ by asserting that for all $A,B \in \mathcal{P}(X)$, we have $R(A,B)$ iff $A$ is not only contained in $B$, but in fact, it can be "swelled" by some non-zero positive number while remaining contained in $B$. See here for a precise definition; the relation of interest is denoted "$\lhd$" in the link.

Answer 1

It would be reasonable to call such things "coreflexive profunctors." Here's why.

The notion of a profunctor comes from category theory. When we specialize to the $\mathrm{Bool}$-enriched case, the following definition is obtained.

Definition 0. Let $P$ and $Q$ denote preordered sets. Then a relation $R : P \nrightarrow Q$ is said to be a ($\mathrm{Bool}$-enriched) profunctor iff the following two axioms hold.

• $p' \lesssim p, R(p,q) \rightarrow R(p',q)$
• $R(p,q), q \lesssim q' \rightarrow R(p,q')$

Now we just have to explain what "coreflexive" means:

Definition 1. Let $P$ denote a preordered set, and $R : P \nrightarrow P$ denote a relation. Then:

• $R$ is reflexive iff $p \lesssim p' \rightarrow R(p,p')$
• $R$ is coreflexive iff $R(p',p) \rightarrow p \lesssim p'$

Note that if $P$ is just a set, then viewing it as a (discrete) poset, the above notion of reflexivity reduces the usual notion.