Equivalent Elements of a Preorder

definition example

In a Preorder $(P,\le )$, elements $x,y$ are equivalent, written $x\cong y$, if $x\le y$ and $y\le x$. This is an Equivalence Relation on $P$ and therefore induces a Partition of $P$ (Example 1.49) — ”$A$ induces $B$” meaning we have an automatic way to turn an $A$ into a $B$.

Sources: 7 Sketches Definition 1.30, Example 1.49, Remark 1.35, 1.82.

A Partial Order is a preorder in which $x\cong y$ implies $x=y$; the quotient $P/\cong$ is always a partial order, the Skeleton of $P$ viewed as a category. Elements defined by universal properties — e.g. two meets of the same subset — are automatically equivalent (Remark 1.82), which is why we may write “the” meet: “any two things defined by the same universal property are unique up to unique isomorphism”. In a category, the analogue of $\cong$ is Isomorphism.