Dense relations

The dense relations (ちゅうみつかんけい British: dense relation) in the mathematics are binary relation R on meeting X, and the thing that R-has a relation with both x and y in former z of X says an existing thing to any 2 yuan x, y in R-relations of X.

If I write it in a sign,

${\displaystyle \forall x\ \forall y\ xRy\Rightarrow (\exists z\ xRz\land zRy)}$

となる.

Any reflection relations are dense.

For example, as for half order of narrow senses <, it is said that it is dense order (dense order) as a binary relation when it is dense as relations. In other words, I say that a thing meeting x <z <y in former z of X exists by all means to a thing meeting x <y in any 2 yuan x, y of X when half order ≤ on meeting X is dense (or ordered set (X,≤)).

The thing which classified the order by normal big things and small things relations into the whole of the rational number is dense in this meaning (as for the ordered set which the whole real number makes, like). On the other hand, the thing which put normal order in the meeting that the whole integer makes is not dense.

Allied item

• クリプキ semantics
• It is dense by oneself
• Dense meeting

References

• David Harel, Dexter Kozen, Jerzy Tiuryn, Dynamic logic, MIT Press, 2000, ISBN 0262082896, p. 6ff

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This article is taken from the Japanese Wikipedia Dense relations

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