Namespaces
Variants
Actions

Difference between revisions of "Elementary interval"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Category:Order, lattices, ordered algebraic structures)
(→‎Comments: more terminology)
 
(2 intermediate revisions by the same user not shown)
Line 2: Line 2:
 
''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.
  
 
$$a\leq x\leq b\Rightarrow a=x\text{ or }a=b.$$
 
$$a\leq x\leq b\Rightarrow a=x\text{ or }a=b.$$
Line 9: Line 9:
  
 
====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]].
 +
 +
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