Associativity and the cosmash product in operadic varieties of algebras

Reimaa, Ülo;Vienne, Corentin;Van der Linden, Tim
(2023) Illinois Journal of Mathematics — Vol. 67, n° 7, p. 563-598 (2023)

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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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Reimaa, Ü., Vienne, C., & Van der Linden, T. (2023). Associativity and the cosmash product in operadic varieties of algebras. Illinois Journal of Mathematics, 67(7), 563-598. https://doi.org/10.1215/00192082-10678862 (Original work published 2023)