TITLE: a reference request to study properties of a set which is closed under a binary operation
QUESTION [1 upvotes]: Let $S$ be a collection of disjoint sets. Let a binary operation '*' defined $\forall x,y$ each belonging to two different sets or a same set in $S$ with the property that $z=x*y$ belongs to some set in $S$ . Let $\forall A_1,A_2 \in S$ a  binary operation $ \otimes$ is defined between $A_1,A_2$ as $B=A_1 \otimes A_2$ as the collection of all $z=(x*y)$ where $x\in A_1,y\in A_2$. Now the set $S$ is closed under the binary operation $\otimes$.
I want to study the properties of such a set $S$ under the operation $\otimes$.My question is what is the subject in mathematics which deals with such a situation ?

REPLY [3 votes]: Sets with a binary operation that is not assumed to be associative are called Magmas (they are sometimes also called "quasigroups", especially in older literature, but that term has a different meaning in category Theory). Your set $S$ with operation $\otimes$ is a magma. 
If the operation happens to be associative, then you have semigroup, though that will depend, in this case, on the binary operation *.
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