Cofinality spectrum problems: The axiomatic approach

The Cofinality Spectrum of the Infinite Symmetric Group

Axiomatic theory of spectrum III. — Semiregularities

The notion of regularity in a Banach algebra was introduced and studied in [KM] and [MM]. A non-empty subset R of a unital Banach algebra A is called a regularity if it satisfies the following two conditions: (i) if a ∈ A and n ∈ N, then a ∈ R ⇔ a ∈ R, (ii) if a, b, c, d are mutually commuting elements of A satisfying ac + bd = 1A then ab ∈ R ⇔ a, b ∈ R. The axioms of regularities are weak enou...

Revisiting Consistency in House Allocation Problems and the Computational Approach to the Axiomatic Method

How should we allocate a social endowment of objects among a group of agents when monetary compensation is not possible? Typical examples are the assignment of offices among faculty and the assignment of tasks among subalterns. We follow an axiomatic approach and impose two properties on rules: Pareto-efficiency and consistency. Previous research arrived at a “dictatorship” result imposing anot...

Cofinality of the Nonstationary Ideal

We show that the reduced cofinality of the nonstationary ideal NSκ on a regular uncountable cardinal κ may be less than its cofinality, where the reduced cofinality of NSκ is the least cardinality of any family F of nonstationary subsets of κ such that every nonstationary subset of κ can be covered by less than κ many members of F . For this we investigate connections of the various cofinalitie...

Axiomatic approach to the cosmological constant

A theory of the cosmological constant Λ is currently out of reach. Still, one can start from a set of axioms that describe the most desirable properties a cosmological constant should have. This can be seen in certain analogy to the Khinchin axioms in information theory, which fix the most desirable properties an information measure should have and that ultimately lead to the Shannon entropy as...