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For a given n the elements of are then called homogeneous elements of degree n. Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree n.
Elements of R that lie inside for some are said to be homogeneous of grade i. The previously defined notion of "graded ring" now becomes the same thing as an N {\displaystyle \mathbb {N} } -graded ring, where N {\displaystyle \mathbb {N} } is the monoid of natural numbers under addition.
If R = k[x 0, ..., x n] (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) is graded with respect to degree, there is a bijective correspondence between projective algebraic sets in projective n-space over k and homogeneous, radical ideals of R not equal to the irrelevant ideal. [2]
Let H be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of I. If I is homogeneous, then H=I. Finally let B be a Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the elements of B.
Homogeneous catalysis, a sequence of chemical reactions that involve a catalyst in the same phase as the reactants Homogeneous (chemistry) , a property of a mixture showing no variation in properties Homogenization (chemistry) , intensive mixing of mutually insoluble substance or groups of substance to obtain a soluble suspension or constant
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
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That is, the maps on X coming from elements of G preserve the structure associated with the category (for example, if X is an object in Diff then the action is required to be by diffeomorphisms). A homogeneous space is a G-space on which G acts transitively. If X is an object of the category C, then the structure of a G-space is a homomorphism: