Let be a partial order. An ideal in is a subset satisfying three properties:
Given , its principal ideal is
This set is clearly nonempty, directed, and a lower set. It is the smallest ideal containing .
Textbook reference:
Davey, B. A., and H. A. Priestley. 2002. Introduction to Lattices and Order. 2nd ed. Cambridge University Press. https://doi.org/10.1017/CBO9780511809088.
@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},
}