Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A (binary) relation $\sim$ on a set $A$ is asymmetric if no two elements are related in both orders:
This is equivalently
In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is asymmetric if it is disjoint from its dual:
Of course, this containment is in fact an equality.
An asymmetric relation is necessarily irreflexive.
That $x \sim y \;\Rightarrow\; y \nsim x$ implies that $x \nsim y \vee y \nsim x$ holds. As a result, the disjunction of $x \sim y$ and $y \sim x$ is equivalent to the exclusive disjunction of $x \sim y$ and $y \sim x$, and is an inequality relation $x \# y$:
If $x \sim y$ is also cotransitive then $x \# y$ is an apartness relation, and if $x \sim y$ is connected then $x \# y$ is tight.
Last revised on September 7, 2024 at 13:29:24. See the history of this page for a list of all contributions to it.