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Difference between revisions of "Elementary interval"

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''of a partially ordered set''
 
''of a partially ordered set''
  
A subset consisting of two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035340/e0353401.png" /> such that there are no other elements in the partially ordered set between them, i.e.
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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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035340/e0353402.png" /></td> </tr></table>
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$$a\leq x\leq b\Rightarrow a=x\text{ or }a=b.$$
  
  
  
 
====Comments====
 
====Comments====
Elementary intervals are also called gaps or atomic intervals.
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Elementary intervals are also called gaps or simple or atomic intervals.
  
For (elementary) intervals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035340/e0353403.png" /> see [[Interval and segment|Interval and segment]]; [[Interval, closed|Interval, closed]]; [[Interval, open|Interval, open]].
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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'' $a$.
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[[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=11924
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article