Name for vector spaces with two algebra structures that satisfy the exchange law Is there a name/reference for the following object? We have a vector space $V$ over some field with two associative bilinear operations $\circ,*:V \times V \to V$ which satisfy the interchange law, i.e.,
$$(a*b)\circ(c*d) = (a\circ c)*(b\circ d) $$
for all $a,b,c,d$ in $V$. My first instinct was to search for "bialgebra" but of course that is an algebra and a coalgebra structure rather than two algebra structures...
 A: If the operations have units ($a * 1 = a$ etc), then that is simply called a commutative algebra: https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_argument Indeed in that case, $a*b = a \circ b = b * a = b \circ a$.
Otherwise, I don't know if there is an established name. This is an algebra over the Boardman-Vogt tensor product of operads $\mathrm{Ass} \otimes_{BV} \mathrm{Ass}$, where $\mathrm{Ass}$ is the operad encoding associative algebras. This is not an uninteresting structure, and it exhibits nontrivial hidden commutativity properties. Here is a summary taken from Boardman--Vogt tensor products of absolutely free operads (Bremner-Dotsenko)

The quoted results are from:

*

*J. Kock: Note on commutativity in double semigroups and two-fold monoidal categories. Journal of Homotopy and Related Structures 2 (2007) no. 2, 217–228.

*M. Bremner, S. Madariaga: Permutation of elements in double semigroups. Semigroup Forum 92 (2016), no. 2, 335–360.

If I had to choose a name, I would call them "double algebras", but I do not believe this is standard. "Double semigroup" certainly is standard terminology, but there is not requirement of linearity.
