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Boardman–Vogt tensor products of absolutely free operads

Part of: Algebraic combinatorics Categories with structure Homological algebra Discrete geometry Combinatorics

Published online by Cambridge University Press:  26 January 2019

Murray Bremner  and
Vladimir Dotsenko
Murray Bremner
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada (bremner@math.usask.ca)
Vladimir Dotsenko
Affiliation:
School of Mathematics, Trinity College Dublin, Ireland and Departamento de Matemáticas, CINVESTAV-IPN, Col. San Pedro Zacatenco, México, D.F., CP 07360, Mexico (vdots@maths.tcd.ie)
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Abstract

To the memory of Trevor Evans (1925–1991),

the pioneer of interchange laws in universal algebra

We establish a combinatorial model for the Boardman–Vogt tensor product of several absolutely free operads, that is, free symmetric operads that are also free as 𝕊-modules. Our results imply that such a tensor product is always a free 𝕊-module, in contrast with the results of Kock and Bremner–Madariaga on hidden commutativity for the Boardman–Vogt tensor square of the operad of non-unital associative algebras.

Keywords

absolutely free operadalgebraic operadinterchange lawnonassociative algebraBoardman–Vogt tensor product of symmetric operadsoperad of little d-rectanglesrectangular partitions of the unit d-dimensional cube

MSC classification

Primary: 18D50: Operads
Secondary: 05A15: Exact enumeration problems, generating functions 05E15: Combinatorial aspects of groups and algebras 18D05: Double categories, $2$-categories, bicategories and generalizations 18G10: Resolutions; derived functors 52C22: Tilings in $n$ dimensions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 367 - 385
Copyright
Copyright © Royal Society of Edinburgh 2019

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