I'm writing notes for my discrete math students and I'm trying to explain transitivity. There's a theorem I want to write down, but I'm not sure if there is a standard name for the object I construct.
Let $R$ be a relation on a set $X$ and let us call a path a finite sequence $x_1x_2 \ldots x_n$ where $x_i$ is related to $x_{i+1}$. If $R$ is transitive then each path has the property that $x_1$ is related to $x_n$.
I'd like to say, "In a transitive relation every path is [blank]." I'm not sure what goes in the blank, if there is standard terminology for a path whose beginning is connected by a directed edge to its end. It's not a circuit, since it's not the case that $x_n$ is related to $x_1$ necessarily. It's not just "closed," is it?
Thanks in advance.
Edit: I apologize if my wording is unclear. I mean to ask if there is a standard name for a directed path $x_1 \to x_2 \to \cdots \to x_n$ where there is also a directed edge $x_1 \to x_n$.
Whatever you do, don't call it closed. Closed paths are already a different thing. If I were writing your notes, I'd say that every path was "shortcutable" ("shortcuttable"?) and try to use that word as infrequently as possible. At least it would help with visualization. Proper graph theory vocabulary will not particularly help you here, because a graph with that property is formally called ... transitive.
(And I for one applaud tying relations to directed graphs. The payoff when you show that the directed graphs of equivalence relations is a union of complete graphs will make it super-obvious how they are related to partitions!)