Monomial order
Definition, details and variations
Definition, details and variations
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Besides respecting multiplication, monomial orders are often required to be [well\-orders](/wiki/Well_ordering "Well ordering"), since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication\-respecting order relations on the set of monomials that are not well\-orders.

In the case of finitely many variables, well\-ordering of a monomial order is equivalent to the conjunction of the following two conditions:
1. The order is a [total order](/wiki/Total_order "Total order").
2. If *u* is any monomial then 1 \\leq u.Since these conditions may be easier to verify for a monomial order defined through an explicit rule, than to directly prove it is a well\-ordering, they are sometimes preferred in definitions of monomial order.

### Leading monomials, terms, and coefficients

The choice of a total order on the monomials allows sorting the terms of a polynomial. The **leading term** of a polynomial is thus the term of the largest monomial (for the chosen monomial ordering). 

Concretely, let  be any ring of polynomials. Then the set  of the (monic) monomials in  is a [basis](/wiki/Basis_%28linear_algebra%29 "Basis (linear algebra)") of , considered as a [vector space](/wiki/Vector_space "Vector space") over the [field](/wiki/Field_%28mathematics%29 "Field (mathematics)") of the coefficients. Thus, any nonzero polynomial  in  has a unique expression 
 p \= \\textstyle\\sum\_{u \\in S} c\_u u  
as a [linear combination](/wiki/Linear_combination "Linear combination") of monomials, where  is a finite subset of  and the  are all nonzero. When a monomial order has been chosen, the **leading monomial** is the largest  in , the **leading coefficient** is the corresponding , and the **leading term** is the corresponding . *Head* monomial/coefficient/term is sometimes used as a synonym of "leading". Some authors use "monomial" instead of "term" and "power product" instead of "monomial". In this article, a monomial is assumed to not include a coefficient.

The defining property of monomial orderings implies that the order of the terms is kept when multiplying a polynomial by a monomial. Also, the leading term of a product of polynomials is the product of the leading terms of the factors.


What are some common applications of non-well-orders on the set of monomials?