# Elementary interval

From Encyclopedia of Mathematics

(Redirected from Atomic interval)

*of a partially ordered set*

A subset consisting of two elements $a\leq b$ such that there are no other elements in the partially ordered set between them, i.e.

$$a\leq x\leq b\Rightarrow a=x\text{ or }a=b.$$

#### Comments

Elementary intervals are also called gaps or simple or atomic intervals.

For (elementary) intervals in $\mathbf R$ see Interval and segment; Interval, closed; Interval, open.

One says that in this situation, $b$ is a *covering element* of, or *covers* $a$.

**How to Cite This Entry:**

Atomic interval.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Atomic_interval&oldid=37630