Often I want to define a structure on a set $S$ which is like a poset, but lacks the antisymmetry condition: i.e., one is allowed both $a\succeq b$ and $a \preceq b$ for $a, b$ different elements of my set. One way to say this is "a category structure with underlying set $S$ which is equivalent to a poset" (where pairs $a, b$ as above are simply related by an isomorphism). But this is a mouthful, and I would like a better, more canonical term for this. Does one exist?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
5
These are called preorders. (The nLab also suggests the term "proset" but I think this is terrible; "proset" should mean a pro-object in sets.) They're the same thing as categories enriched over truth values.
answered Apr 8, 2016 at 16:29
-
2$\begingroup$ Some people also say quasiorder. $\endgroup$ Commented Apr 8, 2016 at 16:54
-
$\begingroup$ I quite dislike both "pre" and "quasi" as prefixes for more general versions of things (especially when they're used to name the more fundamental concept, which I think is the case here), but of the two I prefer "pre" because it's shorter. $\endgroup$ Commented Apr 8, 2016 at 16:56
-
$\begingroup$ I agree. I'm just pointing out both are in use. $\endgroup$ Commented Apr 8, 2016 at 17:43
-
2$\begingroup$ To be fair, the nLab does remark on prosets vs. pro-sets (the latter being used for pro-objects). I'm not sure that the nLab is alone in this particular naming; perhaps it's trying to be descriptive of a usage, not prescriptive in the manner of suggesting that other people use it. But I can make a pretty good guess where it comes from: it is by way of rhyming with "poset" (partially ordered set), "woset" (well-ordered set), "loset" (linearly ordered set) and "toset" (totally ordered set). [I never say "proset" myself.] $\endgroup$ Commented Apr 8, 2016 at 17:45
-
$\begingroup$ So is the pro-proset set empty? $\endgroup$ Commented Apr 8, 2016 at 18:07