If $R$ is a non-transitive binary relation on a set $S$, is there a name for the members of $F \subseteq power(S)$ where:
- $f \in F \text{ and }a,b \in f\ \Rightarrow\ \langle a,b \rangle\, \in R$
- $\bigcup_{f \in F} f = S$
- $f1, f2 \in F \text{ and } f1 \subseteq f2 \Rightarrow f1 = f2$
A relation $R$ on a set $S$ that is reflexive and symmetric but not necessarily transitive is called a "tolerance relation" and makes $(S,R)$ a "tolerance space". Sets that satisfy condition 1 in the OP are ($R$-)preclasses. The maximal ones are ($R$-)classes and they satisfy condition 3 automatically. Together, they together satisfy condition 2.
There are many papers discussing tolerance spaces, but for the classes, Wikipedia cites Tolerance spaces: Origins, theoretical aspects and applications by Peters and Wasilewski.
If the intention was not to assume that $R$ is reflexive, then conditions 1 and 2 may not be satisfied together if $S$ is nonempty. If the intention was to assume reflexivity but not symmetry, then condition 1 forces each $f$ to have a sort of symmetric piece of $S$, and those need not cover $S$ so condition 2 need not be compatible with condition 1.