Definition:Principal Ideal of Preordered Set

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

• Equivalence of Definitions of Principal Ideal of Preordered Set

• 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.

Do not confuse with the word principle, which is a noun.