# a reference request to research buildings of a set which is shut under a binary procedure

Let $S$ be a collection of disjoint collections. Allow a binary procedure' *' specified $\forall x,y$ each coming from 2 various collections or a very same embed in $S$ with the building that $z=x*y$ comes from some embeded in $S$. Allow $\forall A_1,A_2 \in S$ a binary procedure $ \otimes$ is specified in between $A_1,A_2$ as $B=A_1 \otimes A_2$ as the collection of all $z=(x*y)$ where $x\in A_1,y\in A_2$. Currently the set $S$ is shut under the binary procedure $\otimes$.

I intend to research the buildings of such a set $S$ under the procedure $\otimes$. My inquiry is what is the topic in maths which manages such a scenario?

Sets with a binary procedure that is not thought to be associative are called Magmas (they are occasionally additionally called "quasigroups", specifically in older literary works, yet that term has a various definition in group Theory). Your set $S$ with procedure $\otimes$ is a lava.

If the procedure takes place to be associative, after that you have semigroup, though that will certainly depend, in this instance, on the binary procedure *.

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