nLab stable relation

Idea

A relation whose double negation implies the relation itself.

Definition

A stable binary relation $R$ on a set $S$ is a relation such that for all elements $a \in S$ and $b \in S$, $\neg \neg R(a, b)$ implies that $R(a, b)$.

Examples

• Every decidable relation is a stable relation.

• The denial inequality relation of a set is a stable relation.

• In any inequality space, equality is a stable relation.

• Markov's principle states that the pseudo-order for the modulated Cauchy real numbers is a stable relation.

• The analytic Markov's principle states that the pseudo-order for the Dedekind real numbers is a stable relation.

• Given a relation $R$, the negation of the relation $\neg R$ is a stable relation.

• In the presence of the double negation law, every relation is a stable relation.

 See also

• stable proposition

• stable equality

• denial inequality