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How do you call a quantity that is not an invariant, but only changes in one direction during the process?

Example: The degree of the polynomials go down when Euclidean division is applied, so the algorithm stops.

Another example would be entropy.

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  • $\begingroup$ You can also call it non-increasing or non-decreasing, as the case may be. For example, the cumulative distribution function of a random variable is a non-decreasing function. The sequence $1/n!,\ n \ge 0$ is non-increasing. $\endgroup$ – M. Vinay Jun 12 '14 at 12:59
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If your process is continuous, then monotonic increasing (decreasing) is the right word. For example the function graph of $f(x) = 2x + 1$ is monotonic increasing.

In your example with polynomial degrees it is a discrete process an so the word descending (or ascending) is more adequate in my opinion.

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  • $\begingroup$ I really do not think that this is the right word. It would correspond to calling an invariant "constant function". It is not wrong, but it hides the fact that the constantcy serves a particular function in the setup. "Lyapunov function" is not synonymous with "monotonic decreasing function", either, for example. $\endgroup$ – Tara Jun 13 '14 at 5:39
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The term I know is "monovariant," but I mostly only seen this term used in the context of contest math. See, for example, these notes.

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  • $\begingroup$ But my impression is that "monovariant" is not really widely used, even in contests, and it is a "wrong" construction gluing the Greek prefix to the Latin word. It should be univariant if anything, but both words seem to mean "one degree of freedom" which is something very different. $\endgroup$ – Tara Jun 13 '14 at 5:35
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You could say monotonic, or if you want to be more precise, (strictly) decreasing or (strictly) increasing.

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  • $\begingroup$ See my comment to the other similar answer. One would not call a Lyapunov function a decreasing function (although this word might well occur in the definition). $\endgroup$ – Tara Jun 13 '14 at 5:39

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