Quadratic nonsymmetric quaternary operads

TL;DR

This paper investigates quadratic nonsymmetric quaternary operads using computational algebra to analyze relations akin to associativity, focusing on relations with minimal or maximal cubic consequences within algebraic operad theory.

Contribution

It introduces a computational approach to study quadratic nonsymmetric quaternary operads and characterizes relations with extremal cubic consequences.

Findings

Identified relations with minimal cubic consequences

Identified relations with maximal cubic consequences

Applied computational linear algebra to operad theory

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.