Suppose R is a binary relation on a non-empty set S. Let E be an equivalence relation on S. Now form the obvious quotient structure: Let S' be the set of all E-equivalence classes [s] of members s of S, and let R' be the binary relation on S' for which [s]R'[t] if and only if sRt, whenever s and t are in S.
Clearly some properties of R are preserved by taking quotients in this way, while others aren't. For example, transitivity of R entails transitivity of R', but irreflexivity of R does not entail irreflexivity of R'.
Is there a general characterization of the properties of R that are preserved under this quotient construction?
A well defined R' is
[a]R'[b] when exists x in [a], y in [b] with xRy.
If R is reflexive, symmetrical or total,
then R is reflexive, symmetrical or total, respectively.
Other relational properties based upon assuming two instances of the relation require additional assumptions for preservation under quotients.