Ideal (order Theory) - Basic Definitions

Basic Definitions

A non-empty subset I of a partially ordered set (P,≤) is an ideal, if the following conditions hold:

1. For every x in I, y ≤ x implies that y is in I. (I is a lower set)
2. For every x, y in I, there is some element z in I, such that x ≤ z and y ≤ z. (I is a directed set)

While this is the most general way to define an ideal for arbitrary posets, it was originally defined for lattices only. In this case, the following equivalent definition can be given: a subset I of a lattice (P,≤) is an ideal if and only if it is a lower set that is closed under finite joins (suprema), i.e., it is nonempty and for all x, y in I, the element xy of P is also in I.

The dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging with, is a filter. The terms order ideal, order filter, semi-ideal, down-set and decreasing subset are sometimes used for arbitrary lower or upper sets. Wikipedia uses only "ideal/filter (of order theory)" and "lower/upper set" to avoid confusion.

Frink ideals, pseudoideals and Doyle pseudoideals are different generalizations of the notion of a lattice ideal.

An ideal or filter is said to be proper if it is not equal to the whole set P.

The smallest ideal that contains a given element p is a principal ideal and p is said to be a principal element of the ideal in this situation. The principal ideal p for a principal p is thus given by p = {x in P | x ≤ p}.