Nonsymmetric versions of binary quadratic operads

TL;DR

This paper explores nonsymmetric versions of binary quadratic operads, focusing on their construction, properties, and examples within the context of the white Manin product with the associative operad.

Contribution

It introduces the concept of nonsymmetric operads derived from binary quadratic operads and provides criteria and examples for their existence and properties.

Findings

Criteria for when the operad $ ext{As} ullet ext{Var}$ has a nonsymmetric version

Construction of several examples and counterexamples

Description of combinatorial properties of certain nonsymmetric operads

Abstract

In this paper, we study the white Manin product of the associative operad $\As$ with a binary quadratic operad $\Var$. We introduce the notion of a nonsymmetric version of $\Var$ and provide a criterion for determining when the operad $\As\circ\Var$ has this property. We illustrate the construction with several examples and counterexamples. Finally, for some operads admitting nonsymmetric versions, we describe their combinatorial properties.