What do you call a relation that isn't finer than any equivalence, besides the full relation?

Question

In other words, this is a relation whose "equivalence closure" is the full relation. The term "weakly connected" comes to mind naturally, since if you draw a directed graph, with a vertex coming from $x$ to $y$ if and only if $xRy$, then this graph will indeed be weakly connected.

But some sources reserve the term "weakly connected" for relations for which $\forall_x \forall_y xRy \lor y Rx$ holds. From graph-theoretic standpoint, the best name for such relations would be "weakly full", but that's beside the point really.

So, is there a standard solution to this in literature? If not, what can I do?

Answer 1

[Disclaimer: The following is more a non-answer than an answer. I hope someone else can contribute with more insightful ideas.]

We are talking about a binary relation $R \subseteq X \times X$ over some set $X$ such that its equivalence closure $=_R$ (the symmetric and reflexive-transitive closure of $R$, i.e., the intersection of all the equivalence relations on $X$ containing $R$) is the universal (or trivial) relation, that is, $x =_R y$ for all $x,y \in X$ ($=_R$ coincide with $X \times X$).

As far as I know, there is no standard terminology for such a kind of relation $R$. I checked on Wikipedia's pages about binary relations (which contains a glossary their properties) and generated equivalence relations, as well as in Schmidt's Relational Mathematics (a quite encyclopedic textbook about relations), without finding any name.

My proposals: I would say that such a $R$ generates a universal (or trivial) equivalence. If you look for an adjective, I would say that $R$ is coarse or coarse-grained, because the equivalence that it generates is the coarsest equivalence containing $R$. But it depends on the context whether this terminology is appropriate or not. Another proposal is to say that $R$ is (weakly) connecting, because if you represents $R$ as a directed graph (with $X$ as the set of vertices), you get a (weakly) connected graph.

In my opinion, it is misleading to use connex or connected because

• it is quite standard to use connex for a relation $R$ such that $xRy$ or $yRx$ for all $x,y \in X$;
• it is well established that connected refers to (directed) graphs, and the pair $(R,X)$ can be see as a directed graph, but here you are looking for an adjective for $R$ only.

I would not use complete, full or similar either, because this kind of terms is overloaded (think of order theory or graph theory), with different meanings. It is true that if you represent $=_R$ as a graph, then you get a complete graph, but this is a property of $=_R$, not of $R$.