Towards more realistic (e.g., non-associative) and- and oroperations in fuzzy logic

How is fuzzy logic usually formalized? There are many seemingly reasonable require-ments that a logic should satisfy: e.g., since and are the same, the correspond-ing and-operation should be commutative. Sim-ilarly, since means the same as, we should expect that the and-operation should also satisfy this property, etc. It turns out to be im-possible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if the