Gallium trichloride: Difference between revisions

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In [[mathematics]], a [[partial order]] < on a set ''X'' is said to be '''dense''' if, for all ''x'' and ''y'' in ''X'' for which ''x'' < ''y'', there is a ''z'' in ''X'' such that ''x'' < ''z'' < ''y''.
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The [[rational number]]s with the ordinary ordering are a densely ordered set in this sense, as are the [[real number]]s. On the other hand, the ordinary ordering on the [[integer]]s is not dense.
 
==Generalizations==
Any [[binary relation]] ''R'' is said to be ''dense'' if, for all ''R''-related ''x'' and ''y'', there is a ''z'' such that ''x'' and  ''z'' and also ''z'' and  ''y'' are ''R''-related.{{Citation needed|reason=There are reasonable, inequivalent alternatives, so a source should be given.|date=March 2012}} Formally:
: <math> \forall x\  \forall y\ xRy\Rightarrow (\exists z\ xRz \land zRy).</math>
 
Every [[reflexive relation]] is dense. A strict [[partial order]] < is a dense order [[iff]] < is a dense relation.
 
==See also==
*[[Dense set]]
*[[Dense-in-itself]]
*[[Kripke semantics]]
 
== References ==
* [[David Harel]], [[Dexter Kozen]], Jerzy Tiuryn, ''Dynamic logic'', MIT Press, 2000, ISBN 0-262-08289-6, p.&nbsp;6ff
 
[[Category:Mathematical relations]]
[[Category:Order theory]]
 
{{mathlogic-stub}}

Revision as of 01:35, 27 February 2014

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