Prime ideal theorem for weakly dicomplemented lattices

The article continues the formalization of the lattice theory as structures with two binary operations, not in terms of ordering relations. In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding [9] proven some results; we tried however to follow [21], [12], and [13]. All three were devoted to the Stone representation theorem [18] for Boolean or Heyting lattices.

The main aim of the present article was to bridge this gap between general distributive lattices and Boolean algebras, having in mind that the more general approach will eventually replace the common proof of aforementioned articles. Because in Boolean algebras the notions of ultrafilters, prime filters and maximal filters coincide, we decided to construct some concrete examples of ultrafilters in nontrivial Boolean lattice.

In the final section we present Nachbin theorems [15],[1] expressed both in terms of maximal and prime filters and as the unordered spectra of a lattice [11], [10]. This shows that if the notion of maximal and prime filters coincide in the lattice, it is Boolean. Distributive Lattices. University of Missouri Press, Filters - part I.

Formalized Mathematics1 5 Formalized Mathematics5 2 Complete lattices.

Modular Lattice: Definition and examples by Dr Urvashi Arora

Formalized Mathematics2 5 Functions and their basic properties. Formalized Mathematics1 1 : Functions from a set to a set. Formalized Mathematics1 1 Partial functions. Formalized Mathematics1 2 Some basic properties of sets.

Gierz, K. Hofmann, K. Keimel, J. Lawson, M. Mislove, and D. A Compendium of Continuous Lattices. General Lattice Theory.

Boolean prime ideal theorem

Academic Press, New York, Lattice Theory : Foundation. Representation theorem for Heyting lattices. Formalized Mathematics4 1 Homomorphisms of lattices, finite join and finite meet.

Boolean properties of lattices.In mathematicsthe Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma.

Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals of ring theoryor distributive lattices and maximal ideals of order theory. This article focuses on prime ideal theorems from order theory. Although the various prime ideal theorems may appear simple and intuitive, they cannot be deduced in general from the axioms of Zermelo—Fraenkel set theory without the axiom of choice abbreviated ZF.

Instead, some of the statements turn out to be equivalent to the axiom of choice ACwhile others—the Boolean prime ideal theorem, for instance—represent a property that is strictly weaker than AC. An order ideal is a non-empty directed lower set. If the considered partially ordered set poset has binary suprema a. An ideal I is prime if its set-theoretic complement in the poset is a filter.

Ideals are proper if they are not equal to the whole poset. Historically, the first statement relating to later prime ideal theorems was in fact referring to filters—subsets that are ideals with respect to the dual order. The ultrafilter lemma states that every filter on a set is contained within some maximal proper filter—an ultrafilter.

Recall that filters on sets are proper filters of the Boolean algebra of its powerset. In this special case, maximal filters i. The dual of this statement thus assures that every ideal of a powerset is contained in a prime ideal. The above statement led to various generalized prime ideal theorems, each of which exists in a weak and in a strong form.

Weak prime ideal theorems state that every non-trivial algebra of a certain class has at least one prime ideal. In contrast, strong prime ideal theorems require that every ideal that is disjoint from a given filter can be extended to a prime ideal that is still disjoint from that filter.

In the case of algebras that are not posets, one uses different substructures instead of filters. Many forms of these theorems are actually known to be equivalent, so that the assertion that "PIT" holds is usually taken as the assertion that the corresponding statement for Boolean algebras BPI is valid.

Another variation of similar theorems is obtained by replacing each occurrence of prime ideal by maximal ideal. The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras. Thus the formal statement is:. We refer to these statements as the weak and strong BPI. The two are equivalent, as the strong BPI clearly implies the weak BPI, and the reverse implication can be achieved by using the weak BPI to find prime ideals in the appropriate quotient algebra.

This theorem is a well-known fact for Boolean algebras. Its dual establishes the equivalence of prime filters and ultrafilters. Note that the last property is in fact self-dual—only the prior assumption that I is an ideal gives the full characterization. All of the implications within this theorem can be proven in ZF. Note that one requires "global" maximality, not just maximality with respect to being disjoint from F. Yet, this variation yields another equivalent characterization of BPI:.

The fact that this statement is equivalent to BPI is easily established by noting the following theorem: For any distributive lattice Lif an ideal I is maximal among all ideals of L that are disjoint to a given filter Fthen I is a prime ideal. The proof for this statement which can again be carried out in ZF set theory is included in the article on ideals. Since any Boolean algebra is a distributive lattice, this shows the desired implication.

All of the above statements are now easily seen to be equivalent. Going even further, one can exploit the fact the dual orders of Boolean algebras are exactly the Boolean algebras themselves. Hence, when taking the equivalent duals of all former statements, one ends up with a number of theorems that equally apply to Boolean algebras, but where every occurrence of ideal is replaced by filter.

It is worth noting that for the special case where the Boolean algebra under consideration is a powerset with the subset ordering, the "maximal filter theorem" is called the ultrafilter lemma. Summing up, for Boolean algebras, the weak and strong MIT, the weak and strong PIT, and these statements with filters in place of ideals are all equivalent.No tips currently available, please check back later. We also provide Dubai Carnival tips every Thursday and Australian racing tips every evening, updated at 8pm.

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Prime Filters and Ideals in Distributive Lattices

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