Magma

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A magma (or a groupoid) is a set $\displaystyle S$, together with a function $\bot : S \times S \mapsto S$, i.e., a set with a binary operation $\bot$. A set $\displaystyle S$ with an operation $\bot$ that maps some proper subset of $S \times S$ into $\displaystyle S$ may be described as a magma with an operation not everywhere defined on $\displaystyle S$.

Magmas so general that usually one studies special cases of magmas. For example, monoids are associative magmas with an identity.