Binary operation and set of all possible properties I've recently found that there are many binary operation's properties that I heard nothing about them. For example, Super-Commutativity, Poisson identity, Antisymmetry and others.
I guess there is a list of all possible properties that binary operation can have. Am I right? 
Is there some book|paper|online library where I can find what math objects have Super-Commutativity, for example?
 A: I don't think the question is terribly well-defined, and most likely no-one has compiled a big list, because properties people want out of binary operations depend on what they want to do with them. As for why the question is ill-defined, many of the properties you listed require the elements to have some extra structure. Let me illustrate.
Let's suppose we have a binary operation $f: S \times S \to S$, where $f$ is a function and $S$ is a set. Commutativity is easy to define: $f(a, b) = f(b, a)$ for all $a, b \in S$. Similarly, associativity is easy, we have $f(f(a, b), c) = f(a, f(b, c))$.
How about skew-symmetry, like one expects from a Lie bracket or a Poisson bracket. Here, we want that $f(a, b) = -f(b, a)$. But now we run into a problem: there is a minus sign there, and a priori $S$ was just a set. So to define skew-symmetry, we need $S$ to have the structure of an Abelian group.
How about supercommutativity? This says that for $a, b \in S$ homogeneous of degree $|a|$, $|b|$ respectively, that $f(a, b) = (-1)^{|a||b|} f(b, a)$. Notice the $(-1)$ in there, so now we need $S$ to be an abelian group, but we also required that $a, b$ were homogeneous, so $S$ needs to have some grading structure allowing us to decompose arbitrary elements into homogeneous parts.
How about $f$ satisfying the Leibniz rule? So $f(ab, c) = a f(b, c) + b f(a, c)$ (such as one might require on a Poisson bracket). Now I seem to be both multiplying and adding elements of $S$, so $S$ needs to be a ring, or algebra, or something like this.
