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. 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. Some authors use the term ideal to mean a lower set, i.e., they include only condition 2 above. With this weaker definition, an ideal of a lattice seen as a poset is not closed under joins, so it is not necessarily an ideal of the lattice. 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 ↓ {displaystyle downarrow } p for a principal p is thus given by ↓ {displaystyle downarrow } p = {x in P | x ≤ p}. An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order. Such ideals are called prime ideals. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper. For lattices, prime ideals can be characterized as follows: