Equivalent definitions of mathematical structures
https://en.wikipedia.org/wiki/Equivalent_definitions_of_mathematical_structures
In mathematics, equivalent definitions are used in two somewhat different ways.
within a particular mathematical theory, a certain notion may have more than one definition - these definitions are equivalent in the context of a given mathematical structure. Equivalence of two definitions means that a mathematical object satisfies one definition if and only if it satisfies the other definition.
a mathematical structure may have more than one definition; the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object - many different objects may implement the same structure.
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