non-associative operators

Non-associative binary operators can encode more information than associative ones. (Same relationship holds between non-commutative and commutative binary operators too. Both facts stem from the meta principle that introduction of symmetries causes loss of information.) For instance, Henry M. Sheffer provided an alternative, more economic axiomatization of Boolean algebras using his stroke operator |:

This non-associative operator encodes the usual Boolean algebra operators as follows:

Note that a more economic axiomatization may not necessarily be more practical for the working mathematicians. For instance, proving certain identities may now take longer. On the other hand, showing that some structure qualifies to be Boolean algebra may now be easier.