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 $\struct {S, \preceq}$ as a (partial) preorder, calling $\preceq$ a (partial) preorder relation.