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 f : R \times S \rightarrow S and a right external binary operation on ''S'' over ''R'' is a function 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''). Generalizations 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 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 funct ...
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Binary Operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation ''on a set'' is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. An operation of arity two that involves several sets is sometimes also called a ''binary operation''. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may be called simply binary functions. Binary operations are the keystone of most algebraic structures that are studie ...
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