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On the Compactness Property of Mereological Spaces

Polkowski Lech

Fundamenta Informaticae

2020

Vol. 172, nr 1

73--95

Continuing our work on mass-based rough mereologies, we make use of the Stone representation theorem for complete Boolean algebras and we exhibit the existence of a finite base in each mereological space. Those bases in turn allow for the introduction of distributed mereologies; regarding each element of the base as a mereological space, we propose a mechanism for fusing those mereological spaces into a global distributed mereological space. We define distributed mass-assignments and rough inclusions pointing to possible applications.

Remarks on the Stone Spaces of the Integers and the Reals without AC

Herrlich H., Keremedis K., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 2

101--114

In ZF, i.e., the Zermelo–Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2P(X), where 2 is 2 = f0; 1g with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X =ω,R. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

On Scott consequence systems

Dimov G., Vakarelov D.

Fundamenta Informaticae

1998

Vol. 33, nr 1

43-70

The notion of Scott consequence system (briefly, S-system) was introduced by D.Vakarelov in an analogy to a similar notion given by D. Scott. In part one of the paper we study the category Ssyst of all S-systems and all their morphisms. We show that the category DLat of all distributive lattices and all lattice homomorphisms is isomorphic to a reflective full subcategory of the category Ssyst. Extending the representation theory of D. Vakarelo for S-systems in P-systems, we develop an isomorphism theory for S-systems and for Tarski consequence systems. In part two of the paper we prove that the separation theorem for S-systems is equivalent in ZF to some other separation principles, including the separation theorem for filters and ideals in Boolean algebras and separation theorem for convex sets in convexity spaces.