I just read the definition:
Two partial ordered sets X and Y are said to be similar iff there a bijective function from X to Y such that for f(x) < f(y) to occur a necessary and sufficient condition is x < y.
As much as I can understand necessary and sufficient is required because elements may not be comparable and it may occur f(x) < f(y).(this is from book Naive Set Theory by Halmos)
But I also remember that there was same definition for well ordered sets i.e. 'Two well ordered sets X and Y are said to be similar iff there a bijective function from X to Y such that for f(x) < f(y) to occur a necessary and sufficient condition is x < y'
I think we don't need necessary and sufficient condition here. Only necesssary or sufficient condition will be enough. Am I right?
Answer
You are right! If $x
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