Prime Ideal Theorems and systems of finite character

summary:\font\jeden=rsfs7 \font\dva=rsfs10 We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if \text{\jeden S} is a system of finite character then so is the system of all collections of finite subsets of \bigcup \text{\jeden S} meeting a common member of \text{\jeden S}), the Finite Cutset Lemma (a finitary version of the Teichm"uller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma

On calculating residuated approximations and the structure of finite lattices of small width

The concept of a residuated mapping relates to the concept of Galois connections; both arise in the theory of partially ordered sets. They have been applied in mathematical theories (e.g., category theory and formal concept analysis) and in theoretical computer science. The computation of residuated approximations between two lattices is influenced by lattice properties, e.g. distributivity. In previous work, it has been proven that, for any mapping f : L → [special characters omitted] between two complete lattices L and [special characters omitted], there exists a largest residuated mapping ρf dominated by f, and the notion of the shadow σ f of f is introduced. A complete lattice [special characters omitted] is completely distributive if, and only if, the shadow of any mapping f : L → [special characters omitted] from any complete lattice L to [special characters omitted] is residuated. Our objective herein is to study the characterization of the skeleton of a poset and to initiate the creation of a structure theory for finite lattices of small widths. We introduce the notion of the skeleton L˜ of a lattice L and apply it to find a more efficient algorithm to calculate the umbral number for any mapping from a ∼ finite lattice to a complete lattice. We take a maximal autonomous chain containing x as an equivalent class [x] of x. The lattice L˜ is based on the sets {[x] | x ∈ L}. The umbral number for any mapping f : L → [special characters omitted] between two complete lattices is related to the property of L˜. Let L be a lattice satisfying the condition that [x] is finite for all x ∈ L; such an L is called ∼ finite. We define Lo = {[special characters omitted][x] | x ∈ L} and fo = [special characters omitted]. The umbral number for any isotone mapping f is equal to the umbral number for fo, and [special characters omitted] for any ordinal number α. Let [special characters omitted] be the maximal umbral number for all isotone mappings f : L → [special characters omitted] between two complete lattices. If L is a ∼ finite lattice, then [special characters omitted]. The computation of [special characters omitted] is less than or equal to that of [special characters omitted], we have a more efficient method to calculate the umbral number [special characters omitted]. The previous results indicate that the umbral number [special characters omitted] determined by two lattices is determined by their structure, so we want to find out the structure of finite lattices of small widths. We completely determine the structure of lattices of width 2 and initiate a method to illuminate the structure of lattices of larger width