Definition:Preordering/Preordered Set

Definition

Let $S$ be a set.

Let $\precsim$ be a preordering on $S$.

Then the relational structure $\struct {S, \precsim}$ is called a preordered set.

Also known as

This is sometimes shortened to proset.

Some sources use the term quasi-ordering for what we call a preordering, and those sources call this a quasi-ordered set and may shorten it to qoset.

Some sources refer to $\struct {S, \precsim}$ as a (partial) preorder, calling $\precsim$ a (partial) preorder relation.

Also see

• Results about preorderings can be found here.

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Definition $6$