Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.