Ideal

Let (D, \sqsubseteq) be a partial order. An ideal in D is a subset X \subseteq D satisfying three properties:

1. It is nonempty. In other words, X \neq \emptyset.
2. It is a directed set. In other words: x, y \in X \Longrightarrow \exists z \in X.\ x \sqsubseteq z \land y \sqsubseteq z.
3. It is a lower set. In other words: y \sqsubseteq x \in X \Longrightarrow y \in X.

Principal ideals

Given x \in D, its principal ideal is

  \mathop{\downarrow} x = \{ y \in D \mid y \sqsubseteq x \}

This set is clearly nonempty, directed, and a lower set. It is the smallest ideal containing x.

See also

• Ideal completion

References

• PlanetMath: Order Ideal

• Textbook reference:

@book{davey_2002,
	edition = {2},
	title = {Introduction to {Lattices} and {Order}},
	publisher = {Cambridge University Press},
	author = {Davey, B. A. and Priestley, H. A.},
	year = {2002},
	doi = {10.1017/CBO9780511809088},
}