# Difference between revisions of "Elementary interval"

From Encyclopedia of Mathematics

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

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

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====Comments==== | ====Comments==== | ||

− | Elementary intervals are also called gaps or atomic intervals. | + | Elementary intervals are also called gaps or simple or atomic intervals. |

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

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+ | One says that in this situation, $b$ is a ''[[covering element]]'' of, or ''covers'' $a$. | ||

[[Category:Order, lattices, ordered algebraic structures]] | [[Category:Order, lattices, ordered algebraic structures]] |

## Latest revision as of 07:37, 24 January 2016

*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:**

Elementary interval.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Elementary_interval&oldid=33602

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article