A hypermagma is a multivalued map from to , or equivalently, a map from to , or even a magma distributing over both sides of meet and join (such that bottom is absorbing) of a Boolean algebra.
The basic adaptation of associativity in a hypermagma (a hypersemigroup if you will) is to simply say that associativity holds setwise - the set is identical to the set . In the Boolean algebra implementation this is just ordinary associativity.
However, one might feel like this condition, as straightforwardly analogous as it is, to be nonetheless too weak. A hypermagma is said to be coordinated if, for every triple , there exists and such that .
However, it actually unexpectedly turns out that all hypersemigroups are coordinated (PROOF FORTHCOMING).
Coordination is actually strictly weaker than associativity, as evidenced by the hypermagma over a two element set where the only defined product is .
ACTUALLY THIS ENTIRE THING IS SUSPECT AS I CAN ONLY FIND 23 HYPERSEMIGROUPS OF ORDER THREE WHEN THERE SHOULD BE AT LEAST 24