# What to call the initial members of an ordered set?

If I have an ordered set X = {a, b, c} and another ordered set Y = {a, b}, I know that that Y is a subset of X but I also want to convey that Y is the prefix of X if that makes sense. Is there a name for that?

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Y would usually be called an initial segment of X. – Miha Habič Jul 8 '11 at 20:38
@Miha: "Initial segment" is very common for well-ordered sets, less so for partially ordered sets (or for totally ordered sets that don't have a least element). – Arturo Magidin Jul 8 '11 at 20:43

If $(X,\leq)$ is a partially ordered set, then a subset $Y\subseteq X$ is a downward closed subset of $X$ if and only if for all $y\in Y$, if $x\in X$ and $x\leq y$, then $x\in Y$. When $X$ is well-ordered (so that every nonempty subset has a first element), then such a set is usually called an initial segment rather than merely a downward closed set. This is sometimes also used for totally ordered sets with a least element, but not so much for sets that don't have a minimum, since 'initial segment' carries the connotation of a "beginning". Thanks to JDH for pointing out my error of statement here; initial segment is common for any linear order, not only well orders or linear orders with least element.