For a set A with a partially ordering <=, define the following
1) A subset s(x) of A = {y in A such that y <=x}
2) A subset S of A with the property that for every x in S then all y in A which are <= x are also in S.
Most references I find call s(x) the initial segment of the element x in A. But in Azriel Levy’s book
he calls s(x) the initial section of x in A and defines S as an initial segment. I can’t find any other reference to a set defined as S is.
Levy goes on to say that every set s(x) has the property of S (which would appear to follow by transitivity).
To me it would appear that even if the ordering is total then it doesn’t necessarily follow that every S has an x where S = s(x): e.g. A is the rationals with normal ordering and S = {rationals that when squared <= 2} .
I chanced on the “initial section” reference reading a proof of transfinite recursion, p.27 of http://www.uwec.edu/andersrn/SETSIII.pdf
Can anyone clarify segments and sections, confirm common usage of terminology, and give a hint or reference to the usage of sets like S ?
Jean-Louis Krivine in his book on set theory ("Théorie des ensembles") gives the same definition of "initial segment" ("segment initial" in French) as Azriel Levy. Otherwise the phrasings "lower set" and "the set is downward closed" are often used (see http://en.wikipedia.org/wiki/Upper_set).
You say: "even if the ordering is total then it doesn’t necessarily follow that every S has an x where S = s(x)": notice that it might happen even with well-ordered sets (for instance in $\mathbb{R}$). But we have the following: in a well-ordered set $A$, a subset $S$ of $A$ is an initial segment if, and only if, $S = A$ or $S = s(x)$ for some $x \in A$.