  We use computational linear algebra and commutative algebra to study spaces
of relations satisfied by quadrilinear operations. The relations are analogues
of associativity in the sense that they are quadratic (every term involves two
operations) and nonsymmetric (every term involves the identity permutation of
the arguments). We focus on determining those quadratic relations whose cubic
consequences have minimal or maximal rank. We approach these problems from the
point of view of the theory of algebraic operads.
