Definition:Principal Ideal of Preordered Set/Definition 2
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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
- Results about principal ideals of preordered sets can be found here.
Sources
- Mizar article WAYBEL_0:48