Is there a name for magmas with $[x+y]+[x'+y'] \equiv [x+x']+[y+y']$?

Question

Is there a name for magmas (written additively) satsisfying the following identity? The square brackets have no particular signifance, but will hopefully promote readability in what follows. $$[x+y]+[x'+y'] \equiv [x+x']+[y+y'].$$

Example. Any commutative associative magma.

Motivation. Let $A$ and $B$ denote magmas (operations $\oplus$ and $+$ respectively) and assume that $B$ satisfies the above identity. Then for any two homomorphisms $f,g : A \rightarrow B,$ we have that $f+g$ is a homomorphism.

Proof. Consider fixed but arbitrary homomorphisms $f,g : A \rightarrow B$ and suppose $x,y \in A$. Then the following are equal.

1. $(f+g)(x \oplus y)$
2. $f(x \oplus y) + g(x \oplus y)$
3. $[f(x) + f(y)] + [g(x)+g(y)]$
4. $[f(x)+g(x)]+[f(y)+g(y)]$
5. $(f+g)(x) + (f+g)(y).$

We conclude the following. $$(f+g)(x \oplus y) \equiv (f+g)(x) + (f+g)(y).$$

Answer 1

Such a magma is called either medial or entropic.

From lines 1-2 of this paper:

... entropic groupoid (“medial” in the terminology of Ježek-Kepka) ...

refers to Jaroslav Ježek and Tomáš Kepka.

Furthermore, from page 27 of An Introduction to Quasigroups and Their Representations by J.D.H. Smith:

and, from the bibliography of this latter:

• [53] is I.M.H. Etherington Non-Associative Arithmetics, 1949.
• [119] is D.C. Murdoch Structure of abelian quasigroups, 1941.
• [133] is J.H. Przytycki, P. Traczyk Conway algebras and skein equivalences of links, 1987.
• [166] is J.-P. Soublin, Étude algébrique de la notion de moyenne, 1971.
• [168] is S.K. Stein On the foundations of quasigroups, 1957.

Answer 2

These are precisely the commutative magmas in the sense of commutative algebraic theories. They are also known as medial magmas. That the hom sets are again magmas is actually an equivalent characterization. In the language of Durov's thesis we see that commutative magmas are generalized commutative rings.