Ideal (order theory)

In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.

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

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 x${\displaystyle \vee }$y of P is also in I.

The dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging ${\displaystyle \vee }$ with ${\displaystyle \wedge }$, is a filter.

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