What is known about this structure with idempotence, commutativity, cancellation, and another unnamed property?

Question

I'm interested in an algebraic structure $(S,\cdot)$ satisfying the following axioms:

1. Idempotence: $a \cdot a = a$ for all $a \in S$
2. Commutativity: $a \cdot b = b \cdot a$ for all $a, b \in S$
3. Cancellation: if $a \cdot b = a \cdot c$ then $b = c$ for all $a, b, c \in S$
4. Unknown: $(a \cdot b) \cdot (c \cdot d) = (a \cdot c) \cdot (b \cdot d)$ for all $a, b, c, d \in S$

I know of essentially just two examples that aren't also associative:

• Fix a set of variables $\{x_1, \dots, x_n\}$. Define $S$ to be the set of convex combinations of $x_i$ with dyadic rational coefficients, and define $a \cdot b = \frac{a + b}{2}$. This is easy to visualize by letting the variables represent $n$ points that span $(n-1)$-dimensional space with a binary operation that returns the midpoint. We can also relax the coefficients from dyadic rationals to rationals or even reals.
• $S = \{a, b, c\}$ with binary operation $a \cdot b = c$, $a \cdot c = b$, and $b \cdot c = a$

Here's a couple very basic properties I do know:

• From idempotence and (4) we can easily derive $a \cdot (b \cdot c) = (a \cdot b) \cdot (a \cdot c)$, meaning that $\cdot$ distributes over itself.
• From idempotence and cancellation, we can see that $a \cdot b = a$ implies that $b = a$.

I don't know much else about it, and I'm curious if it has a name, other non-associative examples, or any interesting properties. I'm also happy to hear about similar algebraic structures, especially non-associative ones with property (4). There's no application for this exactly, just something I came up with while drawing diagrams related to the first example above.

Answer 1

A few remarks:

• (2) and (4) together imply that $(a_1\cdot a_2)\cdot(a_3\cdot a_4)$ is invariant under permutations of the $a_j$ (and, by induction, similar symmetrically nested products of $2^k$ elements).
• The first construction in the OP (midpoints) also holds in any vector space over a field of odd characteristic (for example, a polynomial ring over $\Bbb F_{233}$).
• The second construction in the OP is a special case of the previous remark, with $a$, $b$, and $c$ being the three elements of $\Bbb F_3$.

Answer 2

The answer to my primary question was given by Arturo Magidin in the comments:

The fourth axiom is sometimes called the "medial law". I think it is sometimes also called "transposition", but this may be less common. A structure satisfying it is called a "medial magma". If it satisfies axioms 2, 3 and 4, it is called a "cancellative commutative medial magma". Finally, if it satisfies axioms 1, 2, 4, and sometimes also 3, it is called a "midpoint algebra". See nLab, this StackExchange answer, and this article ("Internal monoids and groups in the category of commutative cancellative medial magmas" by Fatelo and Martins-Ferreira).