External (mathematics)

The term external is useful for describing certain algebraic structures. The term comes from the concept of an external binary operation which is a binary operation that draws from some external set. To be more specific, a left external binary operation on S over R is a function ${\displaystyle f:R\times S\rightarrow S}$ and a right external binary operation on S over R is a function ${\displaystyle f:S\times R\rightarrow S}$ where S is the set the operation is defined on, and R is the external set (the set the operation is defined over).

The external concept is a generalization rather than a specialization, and as such, it is different from many terms in mathematics. A similar but opposite concept is that of an internal binary function from R to S, defined as a function ${\displaystyle f:R\times R\rightarrow S}$. Internal binary functions are like binary functions, but are a form of specialization, so they only accept a subset of the domains of binary functions. Here we list these terms with the function signatures they imply, along with some examples:

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