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arXiv · 2206.12096

Associativity and the cosmash product in operadic varieties of algebras

Ülo Reimaa·Tim Van der Linden·Corentin Vienne
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Abstract

In this article, we characterise the operadic variety of commutative associative algebras over a field via a (categorical) condition: the associativity of the so-called cosmash product. This condition, which is closely related to commutator theory, is quite strong: for example, groups do not satisfy it. However, in the case of commutative associative algebras, the cosmash product is nothing more than the tensor product; which explains why in this case it is associative. We prove that in the setting of operadic varieties of algebras over a field, it is the only example. Further examples in the non-operadic case are also discussed.

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Ülo Reimaa, Tim Van der Linden, Corentin Vienne. 2022-06-24. Associativity and the cosmash product in operadic varieties of algebras. https://doi.org/10.1215/00192082-10678862

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