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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
up vote 6 down vote accepted

The concept you may be looking for is either an initial segment or downward closed set.

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.

Of course, any finite totally ordered set is well-ordered, so that may be the only case you are interested in, in which case "initial segment" is the common term.

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Arturo, the initial segment terminology is very commonly used for any linear order, whether well-ordered or not. Also, in a partial order, being downward-closed is the same as being an open set in the usual topology having a basis consisting of the lower cones. – JDH Jul 8 '11 at 20:54
@JDH: Thanks for the correction; I've never encountered it except for total orders with least element, and thought it was uncommon; I bow to your superior knowledge here. – Arturo Magidin Jul 8 '11 at 21:02

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