Difference between Ordering and Order?

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I am confused by the two terms order and ordering. I am learning on Ideals, Varieties and Algorithms by Cox et all. The context is monomial orderings and Gröbner basis on polynomial rings.

How are the terms ordering and order different? Are they synonyms?

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2 Answers

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A (total) order is a binary relation in a set $S$ that is reflexive, antisymmetric, transitive and is defined for all pairs of elements.

An ordering is a way to put elements in order. This may be used as a synonym for enumeration, that is, an explicit bijection $\mathbb N \to S$.

For instance, there are several orderings for $\mathbb Q$.

In the context of the book, there are several orderings for the set of monomials in several variables. For instance, you can give priority to $x_1$ or to $x_n$.

On the other hand, Wikipedia says that ordering is the same as total order.

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Order: a binary relation among elements of a set which is: reflexive, transitive, antisymmetric. Ordering: if every empty subset of a set has a minimum. In general, this is the difference.

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