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Difference between revisions of "Elementary interval"

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(Category:Order, lattices, ordered algebraic structures)
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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.
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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.
  
 
$$a\leq x\leq b\Rightarrow a=x\text{ or }a=b.$$
 
$$a\leq x\leq b\Rightarrow a=x\text{ or }a=b.$$
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====Comments====
 
====Comments====
Elementary intervals are also called gaps or atomic intervals.
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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]].
  
 
[[Category:Order, lattices, ordered algebraic structures]]
 
[[Category:Order, lattices, ordered algebraic structures]]

Revision as of 22:04, 12 October 2014

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.

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