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?



What word can I use for a poset with equivalences
These are called [preorders](https://ncatlab.org/nlab/show/preorder). (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. 


