Algebraic structure satisfying interchange law

Question

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 $a, b, c, d \in A$, $$ (a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d).$$

Answer 1

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.