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Suppose we have a non-strict total ordering on tuples of real numbers (the ordering includes tuples of differing lengths). Any tuple, t2, generated from another tuple, t1, by increasing one or more values has t2>t1. Is there necessarily a function f from tuples to real numbers which preserves the ordering? Can this be easily further generalised?

Note:

• By non-strict, I mean that different elements may be equivalent
• This question is actually motivated by investigating Utilitarianism. The tuples are tuples of individual utilities and the aim is to see whether the ability to rank tuples of utilities (with one common sense restriction on the ordering) necessarily means that that a combined utility function can be created
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Suppose we have a non-strict total ordering on tuples of real numbers (the ordering includes tuples of differing lengths) any lengths). Any tuple, t2, generated from another tuple, t1, by increasing one or more values has t2>t1. Is there necessarily a function f from tuples to real numbers which preserves the ordering? Can this be easily further generalised?

I'm not really sure what area of maths this is - combinatorics was my best guess.

Note:

• By non-strict, I mean that different elements may be equivalent
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