Associativity and the cosmash product in operadic varieties of algebras

TL;DR

This paper characterizes commutative associative algebras over a field by a categorical associativity condition on the cosmash product, showing it uniquely identifies this class among operadic varieties of algebras.

Contribution

It establishes that the associativity of the cosmash product characterizes commutative associative algebras in operadic varieties, and discusses non-operadic cases.

Findings

Cosmash product is tensor product for commutative associative algebras.

Associativity of cosmash product uniquely characterizes commutative associative algebras.

Non-operadic cases of cosmash product are also explored.

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.