Reverse mathematics in lattice theory

• Main Author: Ru, Junren
• Other Authors: Wu Guohua
• Format: Theses and Dissertations
• Language: English
• Published: 2018
• Subjects: DRNTU::Science::Mathematics
• Online Access: http://hdl.handle.net/10356/73368
• Institution: Nanyang Technological University

Reverse mathematics is a program of determining which axioms are required to prove theorems of mathematics. This thesis is devoted to the study of reverse mathematics in lattice theory, where several theorems and existence of objects of countable lattices are established from a reverse mathematics point of view. We will mainly focus on the following three subsystems: RCA0, WKL0 and ACA0. In Chapter 2, we first introduce three kinds of ideals in lattices: prime ideals, maximal ideals, relatively maximal ideals, respectively, and then we will investigate the logical strength of the existence of these ideals. Theorem 2.2.5 (RCA0) Every countable lattice (L, , ∨, ∧, 0, 1) contains a maximal ideal. Theorem 2.3.4 (RCA0) Every countable distributive lattice (L, , ∨, ∧, 0, 1) contains a prime ideal. We also study the relations among these ideals over RCA0. Theorem 2.2.6 (RCA0) For a countable lattice (L, , ∨, ∧, 0, 1), (1) Every maximal ideal is relatively maximal, but not conversely. (2) Every prime ideal, if it exists, is relatively maximal, but not conversely. Theorem 2.3.2 (RCA0) For a countable distributive lattice (L, , ∨, ∧, 0, 1), an ideal of L is prime if and only if it is relatively maximal. It is well-known that a lattice L is distributive if and only if neither M3 nor N5 can be embedded into L. It is also known that L is distributive if and only if every relatively maximal ideal in L is prime. We will prove in Chapter 2 that both V statements can be proved in RCA0. In mathematics, a representation theorem states that every abstract structure with certain properties is isomorphic to another structure, which is easy to understand. Because distributivity always holds for the set operations ∪ and ∩, it gave a motivation to construct isomorphisms between distributive lattices and certain spaces of sets. The first such representation theorem is Stone’s Representation Theorem, which states that every Boolean algebra is isomorphic to a certain field of sets. For distributive lattices, we have Birkhoff’s Representation Theorem (for finite distributive lattices) and Priestley’s Representation Theorem (for all bounded distributive lattices). In Chapter 2, we will show that Birkhoff’s Representation Theorem can be proved over RCA0. In the proof of Stone’s Representation Theorem (also in Priestley’s Representation Theorem), the following separation property plays a central role. Lemma 1.5.12 Let L be a distributive lattice, and a, b ∈ L with a b. Then there exists a prime ideal I such that a /∈ I and b ∈ I. In Chapter 3, we will show that the separation property above (for countable distributive lattices) can be proved within WKL0. Actually, we will prove a strong version. Theorem 3.2.1(WKL0) For a countable distributive lattice (L, , ∨, ∧, 0, 1), I an ideal, F a filter of L with I ∩ F = ∅, there exists a prime ideal P containing I but disjoint from F. This strong version is denoted by DPI. For general lattices, DPI is not guaranteed, and we can have a similar separation property, denoted by RMI, by using relatively maximal ideals. We will show in Chapter 4 that RMI can be proved over ACA0. Theorem 4.2.2 (ACA0) In a countable lattice (L, , ∨, ∧, 0, 1), for an ideal I and a filter F of L, if I ∩F = ∅, there exists an ideal M which is maximal among those ideals containing I but disjoint from F. VI In the proof of Birkhoff’s Representation Theorem, the most important component is J (L), the set of join-irreducible elements. In Chapter 4, we will prove that for a countable lattice L, the existence of J (L) is equivalent to ACA0 over RCA0 (Theorem 4.4.1). As a branch of order theory, domain theory studies directed complete partially ordered sets and domains, which formalize the intuitive ideas of approximation and convergence in a general way. In this area, Rudin’s Lemma, a pure result of partially ordered sets, provides many applications in the study of quasi-continuous domains. In Chapter 5, we study Rudin’s Lemma from a reverse mathematics point of view. Theorem 5.3.1 The following are equivalent over RCA0: (1) ACA0; (2) Rudin’s Lemma: Given a countable poset (X, ), and F = hFi : i ∈ Ni a v-directed family of nonempty finite subsets of X, there exists a -directed subset D ⊆ [ i∈N Fi that meets all Fi.