What do we call associative binary operations such that $m(x,m'(y,z)) = m'(m(x,y),z)$?

Question

Question. How do we call a pair of two binary operations $m,m' : X \times X \rightrightarrows X$ on a set such that

1. $m(x,m(y,z)) = m(m(x,y),z)$ ($m$ is associative)
2. $m'(x,m'(y,z)) = m'(m'(x,y),z)$ ($m'$ is associative)
3. $m(x,m'(y,z)) = m'(m(x,y),z)$
4. $m'(x,m(y,z)) = m(m'(x,y),z)$

Maybe we can say "$m$ associates with $m'$" for (3)? But I haven't found this terminology in the literature. Clearly, (4) is symmetric to (3), and an answer to (3) will probably be sufficient. It can also be depicted via a commutative diagram.

$$\require{AMScd}\begin{CD} X \times X \times X @>{X \times m'}>> X \times X \\ @V{m \times X}VV @VV{m}V \\ X \times X @>>{m'}> X \end{CD}$$

Of course the same definitions work in any monoidal category (not necessarily symmetric).

The question A "group" with two binary operations that inter-associate is somewhat related, but it has the additional assumption of a common identity element, which I don't have, and which makes the question kind of trivial. It also suggests the terminology "inter-associate", which couldn't locate in the literature (well, it seems to be used in chemistry for something else).

Notice that $m$ and $m'$ do not necessarily commute, meaning $m(m'(x,y),m'(x',y')) = m'(m(x,x'),m(y,y'))$ does not hold. In fact, the operands $x,y,z$ above always stay in their order.

Background. I was wondering if it's possible to classify the symmetric monoidal structures on the category of sets and found this MO thread. It mentions the very interesting symmetric monoidal structure $$X \otimes_S Y := X \!\times\! S \!\times\! Y + X + Y,$$ where $S$ is any set. The unit object is the empty set $0$. To understand this better, I wanted to determine the monoids with respect to this monoidal structure. For $S = 1$ we just get semigroups (which is kind of funny, semigroups obtain a "neutral element" in this abstract sense, the unique map $0 \to X$). For $S = 2$ we exactly get what I wrote above: two associative operations which "associate with each other". We don't have a neutral element in the classical sense. For a general set $S$, a monoid structure on a set $X$ with respect to $\otimes_S$ consists of binary operations

$$(m_s : X \times X \to X)_{s \in S}$$

such that for all $x,y,z \in X$ and $s,s' \in S$ we have

$$m_s(x,m_{s'}(y,z)) = m_{s'}(m_s(x,y),z))$$

Question. Where can I find more about this interesting symmetric monoidal structure?

Answer 1

As it turns out my advisor Prof. Dr. Birgit Richter has studied this kind of algebraic structure in her diploma thesis (last link on the webpage above) under the German name Doppelalgebra (chapter 3), which translates to double algebra. I do not know, whether this name is standard, or whether this algebraic gadget has been more thoroughly studied by someone before.

When playing around with the obvious generalization to $n$ multiplications, I could not find a name for it in the literature. In my private notes I used $n$-fold (nonunital) algebra, but I do think one can improve on that terminology...

Answer 2

This is a particular case (for $S=\{1,2\}$) of the general algebraic structure that I studied together with Sergey Shadrin and Bruno Vallette for $S$ being any topological space (in this case, one can also take homology and have interesting algebraic structures) in https://arxiv.org/abs/1510.03261, Section 3.1.1. I think we called the corresponding operad $As_S$, and the corresponding algebras $S$-associative algebras.

Answer 3

It has been asked in the comments how $(\mathbf{Set},\otimes_S)$-enriched categories can be described, which is a natural extension of the description of the $(\mathbf{Set},\otimes_S)$-monoids (=one-object enriched categories). Let me answer this here just by unwinding the definitions. Afterwards, we will see that it can be simplified a lot.

A $(\mathbf{Set},\otimes_S)$-enriched category consists of

1. a collection of objects,
2. for every pair of objects $X,Y$ a set $\mathrm{Hom}(X,Y)$,
3. for every $s \in S$ and for every triplet of objects $X,Y,Z$ a map $$\circ_s : \mathrm{Hom}(X,Y) \times \mathrm{Hom}(Y,Z) \to \mathrm{Hom}(Y,Z)$$
4. for every triplet of objects $X,Y,Z$ a map $$\rho_{X,Y,Z} : \mathrm{Hom}(X,Y) \to \mathrm{Hom}(X,Z)$$
5. for every triplet of objects $X,Y,Z$ a map $$\lambda_{X,Y,Z} : \mathrm{Hom}(Y,Z) \to \mathrm{Hom}(X,Z)$$

subject to the following axioms:

1. Right neutrality: $\rho_{X,Y,Y} = \mathrm{id}$
2. Left neutrality: $\lambda_{Y,Y,Z} = \mathrm{id}$
3. Coherence of $\rho$ with $\rho$: $$\begin{array}{ccc} \mathrm{Hom}(X,Y) & \xrightarrow{\rho} & \mathrm{Hom}(X,U) \\ \hspace{2cm} {\scriptstyle \rho} \searrow & & \hspace{-2cm} \nearrow {\scriptstyle \rho} \\ & \mathrm{Hom}(X,Z) & \end{array}$$
4. Coherence of $\lambda$ with $\lambda$: $$\begin{array}{ccc} \mathrm{Hom}(Z,U) & \xrightarrow{\lambda} & \mathrm{Hom}(X,U) \\ \hspace{2cm} {\scriptstyle \lambda} \searrow & & \hspace{-2cm} \nearrow {\scriptstyle \lambda} \\ & \mathrm{Hom}(Y,Z) & \end{array}$$
5. Coherence of $\lambda$ with $\rho$: $$\require{AMScd}\begin{CD} \mathrm{Hom}(Y,Z) @>{\rho}>> \mathrm{Hom}(Y,U) \\ @V{\lambda}VV @VV{\lambda}V \\ \mathrm{Hom}(X,Z) @>>{\rho}> \mathrm{Hom}(X,U) \end{CD}$$
6. Coherence of $\rho$ with $\circ_s$ for all $s \in S$: $$\begin{CD} \mathrm{Hom}(X,Y) \times \mathrm{Hom}(Y,Z) @>{\mathrm{id} \times \rho}>> \mathrm{Hom}(X,Y) \times \mathrm{Hom}(Y,U) \\ @V{\circ_s}VV @VV{\circ_s}V \\ \mathrm{Hom}(X,Z) @>>{\rho}> \mathrm{Hom}(X,U) \end{CD}$$
7. Coherence of $\lambda$ with $\circ_s$ for all $s \in S$: $$\begin{CD} \mathrm{Hom}(Y,Z) \times \mathrm{Hom}(Z,U) @>{\lambda \times \mathrm{id}}>> \mathrm{Hom}(X,Z) \times \mathrm{Hom}(Z,U) \\ @V{\circ_s}VV @VV{\circ_s}V \\ \mathrm{Hom}(Y,U) @>>{\lambda}> \mathrm{Hom}(X,U) \end{CD}$$
8. Coherence of $\circ_s,\lambda,\rho$ for all $s \in S$: $$\begin{CD} \mathrm{Hom}(X,Y) \times \mathrm{Hom}(Z,U) @>{\mathrm{id} \times \lambda}>> \mathrm{Hom}(X,Y) \times \mathrm{Hom}(Y,U) \\ @V{\rho \times \mathrm{id}}VV @VV{\circ_s}V \\ \mathrm{Hom}(X,Z) \times \mathrm{Hom}(Z,U) @>>{\circ_s}> \mathrm{Hom}(X,U) \end{CD}$$
9. Coherence of $\circ_s$ with $\circ_{s'}$ for all $s,s' \in S$: $$\begin{CD} \mathrm{Hom}(X,Y) \times \mathrm{Hom}(Y,Z) \times \mathrm{Hom}(Z,U) @>{\mathrm{id} \times \circ_{s'}}>> \mathrm{Hom}(X,Y) \times \mathrm{Hom}(Y,U) \\ @V{\circ_s \times \mathrm{id}}VV @VV{\circ_s}V \\ \mathrm{Hom}(X,Z) \times \mathrm{Hom}(Z,U) @>>{\circ_{s'}}> \mathrm{Hom}(X,U) \end{CD}$$

It follows from 9. in particular that each $\circ_s$ is associative, i.e. we have a bunch of semicategory structures. The axiom 9. is a natural way of saying that they are compatible with each other.

Axioms 1 and 3 imply that each $\rho$ is an isomorphism. Likewise, axioms 2 and 4 imply that each $\lambda$ is an isomorphism. So we are left with $\mathrm{Hom}(X,X)$, which becomes a $(\mathbf{Set},\otimes_S)$-monoid because of axiom 9, but they are all isomorphic because of axioms 6 and 7. I guess that this implies that the category of small $(\mathbf{Set},\otimes_S)$-enriched categories with at least one object is equivalent to $\mathbf{Set}_{\neq \emptyset} \times \mathrm{Mon}(\mathbf{Set},\otimes_S)$.