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Abstract
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.
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Notes
No potential conflict of interest was reported by the authors.
Additional information
Funding
Murray Bremner was supported by a Discovery Grant from NSERC, the Natural Sciences and Engineering Research Council of Canada. His visit to Cape Town in November–December 2015 was supported by a grant from NRF, the National Research Foundation of South Africa.. He thanks the Department of Mathematics and Applied Mathematics at the University of Cape Town for its hospitality.