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Compactness and Löwenheim-Skolem properties in categories of pre-institutions

Antonino Salibra, Giuseppe Scollo (1993)

Banach Center Publications

The abstract model-theoretic concepts of compactness and Löwenheim-Skolem properties are investigated in the "softer" framework of pre-institutions [18]. Two compactness results are presented in this paper: a more informative reformulation of the compactness theorem for pre-institution transformations, and a theorem on natural equivalences with an abstract form of the first-order pre-institution. These results rely on notions of compact transformation, which are introduced as arrow-oriented generalizations...

Compactness in Metric Spaces

Kazuhisa Nakasho, Keiko Narita, Yasunari Shidama (2016)

Formalized Mathematics

In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness,...

Compactness of Powers of ω

Paolo Lipparini (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.

Comparing the closed almost disjointness and dominating numbers

Dilip Raghavan, Saharon Shelah (2012)

Fundamenta Mathematicae

We prove that if there is a dominating family of size ℵ₁, then there are ℵ₁ many compact subsets of ω ω whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

Comparing the succinctness of monadic query languages over finite trees

Martin Grohe, Nicole Schweikardt (2004)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study the succinctness of monadic second-order logic and a variety of monadic fixed point logics on trees. All these languages are known to have the same expressive power on trees, but some can express the same queries much more succinctly than others. For example, we show that, under some complexity theoretic assumption, monadic second-order logic is non-elementarily more succinct than monadic least fixed point logic, which in turn is non-elementarily more succinct than monadic datalog. Succinctness...

Comparing the succinctness of monadic query languages over finite trees

Martin Grohe, Nicole Schweikardt (2010)

RAIRO - Theoretical Informatics and Applications

We study the succinctness of monadic second-order logic and a variety of monadic fixed point logics on trees. All these languages are known to have the same expressive power on trees, but some can express the same queries much more succinctly than others. For example, we show that, under some complexity theoretic assumption, monadic second-order logic is non-elementarily more succinct than monadic least fixed point logic, which in turn is non-elementarily more succinct than monadic datalog.
Succinctness...

Comparison game on Borel ideals

Michael Hrušák, David Meza-Alcántara (2011)

Commentationes Mathematicae Universitatis Carolinae

We propose and study a “classification” of Borel ideals based on a natural infinite game involving a pair of ideals. The game induces a pre-order ⊑ and the corresponding equivalence relation. The pre-order is well founded and “almost linear”. We concentrate on F σ and F σ δ ideals. In particular, we show that all F σ -ideals are ⊑ -equivalent and form the least equivalence class. There is also a least class of non- F σ Borel ideals, and there are at least two distinct classes of F σ δ non- F σ ideals.

Compatibility and central elements in pseudo-effect algebras

Paolo Vitolo (2010)

Kybernetika

An equivalent definition of compatibility in pseudo-effect algebras is given, and its relationships with central elements are investigated. Furthermore, pseudo-MV-algebras are characterized among pseudo-effect algebras by means of compatibility.

Compatible Idempotent Terms in Universal Algebra

Ivan Chajda, Antonio Ledda, Francesco Paoli (2014)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a “well-behaved core”, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this “core” is a retract determined by an idempotent endomorphism that is uniformly term definable (through a unary term t ( x ) ) in every member of the given variety. Here, we try to give a unified account of this phenomenon....

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