Gr\"obner Bases for Coloured Operads
Vladislav Kharitonov, Anton Khoroshkin
TL;DR
This paper develops a framework for Gr"obner bases in coloured operads, defining them as monoids in monoidal categories, and demonstrates their application to the Lie-Rinehart operad, which models functions and vector fields.
Contribution
It introduces a new approach to Gr"obner bases for coloured operads, expanding algebraic tools for operad theory and its applications.
Findings
Established a definition of coloured operads as monoids in monoidal categories.
Developed machinery for Gr"obner bases in coloured operads.
Proved the existence of quadratic Gr"obner bases for key examples like the Lie-Rinehart operad.
Abstract
In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Gr\"obner bases for coloured operads. Among the examples for which we show the existance of a quadratic Gr\"obner basis we consider the seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).
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