Order type relation in poset and well ordered sets 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?

You are right! If $x<y$ implies $f(x)<f(y)$, then $x\ge y$ implies either $y<x$, in which case $f(y)<f(x)$, or $x=y$, so that $f(x)=f(y)$. Either way, $x\not< y$ implies $f(x)\not<f(y)$, so that sufficiency implies necessity. A symmetric argument works for necessity implying sufficiency. This holds for any totally ordered set.
