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In mathematics, a monomial order is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the following two properties: # If and ''w'' is any other monomial, then . In other words, the ordering respects multiplication. # If ''u'' is any monomial then . These conditions imply that * If ''u'' and ''v'' are any monomials then . They imply also that the ordering is a well ordering, which means that every strictly decreasing sequence of monomials is finite, or equivalently that every non-empty set of monomials has a minimal element. Among the powers of any one variable ''x'', the only ordering satisfying these conditions is the natural ordering 1<''x''<x2<x3... (with only the first condition, the opposite ordering would also qualify, but the set of all powers of ''x'' would fail to have a minimal element). Therefore the notion of monomial ordering is interesting only in the case of multiple variables. Monomial orderings are most commonly used with Gröbner bases and multivariate division. == Examples == The monomial order implies an order on the individual indeterminates. One can simplify the classification of monomial orders by assuming that the indeterminates are named ''x''1, ''x''2, ''x''3, ... in decreasing order for the monomial order considered, so that always . (If there should be infinitely many indeterminates, this convention is incompatible with the condition of being a well ordering, and one would be forced to use the opposite ordering; however the case of polynomials in infinitely many variables is rarely considered.) In the example below we shall use ''x'' instead of ''x''1, ''y'' instead of ''x''2, and ''z'' instead of ''x''3. With this convention there are still many examples of different monomial orders. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Monomial order」の詳細全文を読む スポンサード リンク
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