Ring-like strusture with non associative addition

There is this structure i found which is the set of continuous maps from [-1, 1]^n into itself, endowed with a "sum" which is the pointwise sum of two functions divided by 2, and a "product" which is the pointwise multiplication of two functions, where the product vector is obtained by component-wise multiplication.

I found that:

sum IS NOT associative

sum IS commutative

a + a = a (sum is idempotent)

for all a there exists a unique b such that a + b = 0

NB: by zero I meant the constant null function, not addition's neutral element (which doesn't exist). This property is meaningful cause it only holds for zero.

prpduct is associative

product is commutative

it's got a unit, the constant (1,1,...,1) function, which is the only invertible element together with the other combinations of 1's and -1's stuffed in a vector product distributes over addition

Question is: has this got any name? Are there any publications dealing with similar structures? I found some on nonassociative rings (which maybe are called somethig else), i mean ring with non associative multiplication, but I didn't find anything about structures like this.

Post scriptum: there could also be an additional operation: composition.