Реттік жиынтар мен сүзгілер: теория және қолданыстары

Nonempty, upper bounded, downward closed subset

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.

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 ideal I is a if it is proper and there is no proper ideal J that is a strict superset of I. Likewise, a filter F is maximal if it is proper and there is no proper filter that is a strict superset. When a poset is a distributive lattice, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general. Maximal filters are sometimes called ultrafilters, but this terminology is often reserved for Boolean algebras, where a maximal filter (ideal) is a filter (ideal) that contains exactly one of the elements {a, ¬a}, for each element a of the Boolean algebra. In Boolean algebras, the terms prime ideal and maximal ideal coincide, as do the terms prime filter and maximal filter. There is another interesting notion of maximality of ideals: Consider an ideal I and a filter F such that I is disjoint from F. We are interested in an ideal M that is maximal among all ideals that contain I and are disjoint from F. In the case of distributive lattices such an M is always a prime ideal. A proof of this statement follows. However, in general it is not clear whether there exists any ideal M that is maximal in this sense. Yet, if we assume the axiom of choice in our set theory, then the existence of M for every disjoint filter–ideal pair can be shown. In the special case that the considered order is a Boolean algebra, this theorem is called the Boolean prime ideal theorem. It is strictly weaker than the axiom of choice and it turns out that nothing more is needed for many order theoretic applications of ideals.

The construction of ideals and filters is an important tool in many applications of order theory. In Stone's representation theorem for Boolean algebras, the maximal ideals (or, equivalently via the negation map, ultrafilters) are used to obtain the set of points of a topological space, whose clopen sets are isomorphic to the original Boolean algebra. Order theory knows many completion procedures to turn posets into posets with additional completeness properties. For example, the ideal completion of a given partial order P is the set of all ideals of P ordered by subset inclusion. This construction yields the free dcpo generated by P. An ideal is principal if and only if it is compact in the ideal completion, so the original poset can be recovered as the sub poset consisting of compact elements. Furthermore, every algebraic dcpo can be reconstructed as the ideal completion of its set of compact elements.

Ideals were introduced by Marshall H. Stone first for Boolean algebras, where the name was derived from the ring ideals of abstract algebra. He adopted this terminology because, using the isomorphism of the categories of Boolean algebras and of Boolean rings, the two notions do indeed coincide. Generalization to any posets was done by Frink.