Definition:Principal Ideal of Preordered Set
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Definition
Let $\struct {S, \preceq}$ be a preordered set.
Let $I$ be an ideal in $S$.
Definition 1
Then $I$ is a principal ideal if and only if:
- $\exists x \in I: x$ is upper bound for $I$
Definition 2
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.
Linguistic Note
The word principal is (except in the context of economics) an adjective which means main.