Definition:Principal Ideal of Preordered Set/Definition 2

Definition

Let $\struct {S, \preceq}$ be a preordered set.

Let $I$ be an ideal in $S$.

Then $I$ is a principal ideal if and only if:

$\exists x \in S: I = x^\preceq$

where $x^\preceq$ denotes the lower closure of $x$.

Also see

• Equivalence of Definitions of Principal Ideal of Preordered Set

• Results about principal ideals of preordered sets can be found here.

Sources

• Mizar article WAYBEL_0:48