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Is there a name for an algebraic structure that satisfies a rule equivalent to the interchange law from category theory? Say, a set $A$ with associative (not necessarily commutative) operations $\cdot$ and $\circ$ such that for $a, b, c, d \in A$, $$ (a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d).$$

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2 Answers 2

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A duoid comprises a pair of monoids structures $(A, \circ)$ and $(A, \cdot)$ on the same set satisfying the interchange law $(a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d)$. For instance, see (2.4) in Garner–Franco's Commutativity for a definition valid in any duoidal category.

However, note that a duoid in $\mathbf{Set}$ is simply a commutative monoid by the Eckmann–Hilton argument.

See James Francese's answer for a notion of algebra satisfying the interchange law that is not trivial in $\mathbf{Set}$.

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  • $\begingroup$ Does horizontal and vertical composition of natural transformation apply here? $\endgroup$
    – PinkRabbit
    Apr 18 at 9:00
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The term you’d want is interchange algebra, for example as listed in the Encyclopedia of Types of Algebras, although no citations are given there — however it does make sure to distinguish this from a duoid, since the two operations are not necessarily monoids. This way you avoid the Eckmann-Hilton argument and non-trivial examples arise — see for instance this paper, which also cites the “Encyclopedia” (written by Loday under a pseudonym by the way).

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