Is there a binary operator (besides composition) closed under permutations or a notion of a metric space on permutations? When i say "a binary operator closed under permutations" I mean, given $2$ (finite, same number of elements) permutations $p_1$, $p_2$ , is there an operator "$+$" such that $p_1+p_2=p_3$ ($p_3$ a permutation in the same space). If there is an even "stronger" structure, like a ring on permutations it would also be awesome to know.
The second part of the question is asking for a notion of distance between permutations.
Extra Notes:
 I'm experimenting with operators of permutations in stochastic optimization algorithms. It would be useful to find an additive, closed invertible operation on permutations.
 A: Binary operations
It would be more standard to call what you are describing a "binary operator on (a set of) permutations." And likewise, you'd say a "ring of permutations" or "a ring structure on permutations."
As has already been mentioned, composition is the most natural binary operation on permutations. The full set of permutations is a group under this operation, although you probably wouldn't write it additively since it is not a commutative operation.
No natural commutative operations come to mind, and I have not encountered any ring structure on permutations before.
Distance function
You might have better luck coming up with distance functions on permutations, though, at least for permutations on finite sets. Suppose the permutations are represented in the following way. First, enumerate the elements of the set. We'll just go ahead and confuse the elements for their labels so they look like 0, 1, 2, etc. Represent the permutation as a vector which lists the images in this order. So if 0 goes to 2, 1 goes to 5 and 3 goes to 3, the vector would start $(2, 5, 3\ldots)$. Then one metric would be to count the number of places two vectors differ.
The metric would still work for permutations on an infinite set as long as all the permutations you are using each only move finitely many elements.
