For example, $(A\subseteq B$ and $B\subseteq A)$ can be trivially reduced to $(A=B)$. Or $(A\subseteq B$, $B\subseteq C$, and $C\subseteq A)$ can be reduced to equality of all three.
But is it always possible to for any system of relations to reduce it to some "canonical" form, whatever that may be? Such that if there are two equivalent systems, they'd reduce to the same form? If so, is there an efficient algorithm to do so?
Alternatively, what I really want to know, is there any algorithm to determine whether two systems of relations, e.g. $(A\subseteq B\subseteq C\subseteq A)$ and $(A\supseteq B\supseteq C\supseteq A)$ are equivalent to each other? The only operations I'm concerned about are sub/superset (not proper), intersection, and union.