# Definition of similarity mapping between ordered sets: why is a “ strictly precedes” relation required on each side of the biconditionnal?

As a definition of " similarity mapping" I read in Lipschutz, Set Theory :

the mapping f from A to B ( A and B being ordered sets) is a similarity mapping iff, (a) f is a bijection and (b) for any elements a and a' belonging to A :

                   a < a'  iff f(a) < f(a')


The definition is expressed in terms of < : "strictly precedes".

Would f also be a similarity mapping in case < were replaced by " precedes or is equal to" ?

What I do not understand is that (1) the author has defined ordered sets in terms of " precedes or is equal to " and (2) defines an " order preserving function" in terms of " strictly precedes".

Yes, the two notations are equivalent because $$f$$ is also required to be a bijection. This condition ensures that $$f(a) = f(a') \iff a = a'$$.
The given condition implies $$a\le a'\implies f(a) \le f(a')$$, but not the other way around, as any constant function is a counterexample.
• $f$ is also a bijection. – Lord_Farin May 4 at 12:31