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The definitions given above are all specialized cases of the following more general notion of homogeneity in which <math>X</math> can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of a [[monoid]].
Let <math>M</math> be a [[monoid]] with identity element <math>1 \in M,</math> let <math>X</math> and <math>Y</math> be sets, and suppose that on both <math>X</math> and <math>Y</math> there are defined monoid actions of <math>M.</math> Let <math>k</math> be a non-negative integer and let <math>f : X \to Y</math> be a map. Then <math>f</math> is said to be
<math display="block">f(mx) = m^k f(x).</math>
If in addition there is a function <math>M \to M,</math> denoted by <math>m \mapsto |m|,</math> called an {{em|[[absolute value]]}} then <math>f</math> is said to be
<math display="block">f(mx) = |m|^k f(x).</math>
A function is
More generally, it is possible for the symbols <math>m^k</math> to be defined for <math>m \in M</math> with <math>k</math> being something other than an integer (for example, if <math>M</math> is the real numbers and <math>k</math> is a non-zero real number then <math>m^k</math> is defined even though <math>k</math> is not an integer). If this is the case then <math>f</math> will be called
<math display="block">f(mx) = m^k f(x) \quad \text{ for every } x \in X \text{ and } m \in M.</math>
The notion of being
===Distributions (generalized functions)===
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