On the number of distributive lattices.
Struktur- und Anzahlformeln für Topologien auf endlichen Mengen.
-completeness and fixpoint properties
A primrose path from Krull to Zorn
Given a set of “indeterminates” and a field , an ideal in the polynomial ring is called conservative if it contains with any polynomial all of its monomials. The map yields an isomorphism between the power set and the complete lattice of all conservative prime ideals of . Moreover, the members of any system of finite character are in one-to-one correspondence with the conservative prime ideals contained in , and the maximal members of correspond to the maximal ideals contained in...
Prime Ideal Theorems and systems of finite character
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 is a system of finite character then so is the system of all collections of finite subsets of meeting a common member of ), the Finite Cutset Lemma (a finitary version of the Teichm“uller-Tukey Lemma), and various compactness theorems. Several implications between these statements...
Prime and maximal ideals of partially ordered sets
-distributive function lattices
It is known that for a nonempty topological space and a nonsingleton complete lattice endowed with the Scott topology, the partially ordered set of all continuous functions from into is a continuous lattice if and only if both and the open set lattice are continuous lattices. This result extends to certain classes of -distributive lattices, where is a subset system replacing the system of all directed subsets (for which the -distributive complete lattices are just the continuous...
Natural continuity space structures on dual Heyting algebras
Uniform ideal completions
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