In algebra, we study sets with binary operations satisfying certain properties. However, sometimes that the calculation is undefined for some ordered pairs, such as $a\div 0$ for any complex number $a$.
My question is, do people study algebra systems with an operation which is similar to binary operation but allowing the operation to be undefined for some ordered pairs? That is, give a set $X$, are people interested in maps from a subset of $X\times X\to X$ satisfying some properties such as associativity and commutativity (when the product is defined)?