On the cardinality and weight spectra of compact spaces, II

带基数比较的三段论逻辑的有限谱

Finite Spectra of Syllogistic Logic with Cardinality Comparisons

本文对带基数比较的三段论逻辑$S^\dagger(card)$的有限谱给出完全分类，并证明加入句子级布尔连接词后，其谱集构成由原谱生成的拓扑。

Internal sizes in $μ$-abstract elementary classes

Working in the context of $μ$-abstract elementary classes ($μ$-AECs) - or, equivalently, accessible categories with all morphisms monomorphisms - we examine the two natural notions of size that occur, namely cardinality of underlying sets and internal size. The latter, purely category-theoretic, notion generalizes e.g. density character in complete metric spaces and cardinality of orthogonal bases in Hilbert spaces. We consider the relationship between these notions under mild set-theoretic hypotheses, including weakenings of the singular cardinal hypothesis. We also establish preliminary resu…

Spectral spaces of countable abelian lattice-ordered groups

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian {\ell}-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countabl…