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 is an ideal, if the following conditions hold: I is non-empty, for every x in I and y in P, y ≤ x implies that y is in I (I is a lower set), 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 is an ideal if and only if it is a lower set that is closed under finite joins (suprema); that is, it is nonempty and for all x, y in I, the element of P is also in I. A weaker notion of order ideal is defined to be a subset of a poset P that satisfies the above conditions 1 and 2. In other words, an order ideal is simply a lower set. Similarly, an ideal can also be defined as a "directed lower set". The dual notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging with is a filter. 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 and p is said to be a of the ideal in this situation. The principal ideal for a principal p is thus given by ↓ p = . The above definitions of "ideal" and "order ideal" are the standard ones, but there is some confusion in terminology. Sometimes the words and definitions such as "ideal", "order ideal", "Frink ideal", or "partial order ideal" mean one another. An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.