Is there a known example of an algebra $(A, +, \cdot)$ with two binary commutative (see P.S below) and idempotent operations $+$ and $\cdot$ satisfying the identity $(a+b)(c+d)=ac+bd$? 




---


Actually I need to know is there any algebra $A$ having signature $\mathcal{L}$ such that for any $n$-ary $f\in \mathcal{L}$ and any $m$-ary $g\in\mathcal{L}$ we have 
$$
f(g(x\_{11}, \ldots, x\_{1m}), \ldots, g(x\_{n1}, \ldots,x\_{nm}))=g(f(x\_{11}, \ldots,x\_{n1}), \ldots, f(x\_{1m},\ldots, x\_{nm}))
$$
Also I need to have all operations idempotent. As my motivation, I should say that for any variety of such algebras, we have the following interesting property: Let $F=F\_V(x\_1,\ldots,x\_n)$ be a relative free algebra in $V$. Then the solution set of any system of equations in $F$ is a subalgebra of $F^k$ where $k$ is the number of indeterminate. 


P.S. Instead of commutativity, consider the next two weaker conditions:


1. $(a+b)+(c+d)=(a+c)+(b+d)$.
2. $(ab)(cd)=(ac)(bd)$.
\


P.S.2. It is good to use the phrase Medial instead of Commutative. So my question becomes: Is there a known example of an algebra $(A, +, \cdot)$ with two binary medial and idempotent operations satisfying the identity $(a+b)(c+d)=ac+bd$?



Examples of algebras satisfying (a+b)(c+d)=ac+bd
Assuming the properties of idempotency and commutativity, $a+b=(a+b)(a+b)=(a+b)(b+a)=ab+ba=ab+ab=ab$. So you will have essentially identical operations for + and \*. I don't know what happens if you drop one or more of idempotency or commutativity. 


