Homogeneous function: Difference between revisions

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=== Homogeneity under a monoid action ===

The definitions given above are all specializesspecialized 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 '''homogeneous of degree <math>k</math> over <math>M</math>''' if for every <math>x \in X</math> and <math>m \in M,</math>