# 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...

• If we ignore the requirement that V is a vector space and $\circ,*$ are associative and bilinear, then this question gives some examples of such structures: mathoverflow.net/q/154550/22277 Nov 23 at 21:42
• There's a related notion in Definition 1 of my paper arxiv.org/pdf/0912.5307.pdf Nov 29 at 11:06
• Thanks, @AndréHenriques! Serves me right for asking here before walking over to ask you or Chris :) In my case there is no unital structure to the two algebras, so I am saved from Eckmann Hilton through other means... Nov 29 at 13:14

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)
• I will confess that I had to read Prop 1.6 about 5 times before I realized that the two sides of the equality were actually different! For others suffering similarly, note that $x_6$ and $x_7$ are swapped. (And yes, I was indeed aware of Eckmann-Hilton but none of my products are unital). I will wait a bit to see if anyone else has seen the structure and given it a catchy name, and will happily accept this answer otherwise :) Nov 24 at 0:42