Abstract simplicial complexes (here briefly complexes) are set systems on an arbitrary set which are object of study in many areas of both mathematics and theoretical computer science. Usually, they are investigated over finite sets. However, in general, when we consider an arbitrary set Ω (not necessarily finite) and a complex C on Ω, the most natural property related to finiteness is the following: for any subset X of Ω, if F ∈ C for all finite subsets F of X, then X ∈ C. We call locally finite any complex C having such a property. Bearing in mind some motivations and constructions derived from the analysis of information systems in rough set theory, in this paper we associate with any locally finite complex C a corresponding pre- closure operator σC and, through it, we establish several properties of C. Next, we investigate the main features of the specific sub-class of locally finite complexes C for which σC is a closure operator. We call these complexes closable and exhibit a particular family of closable locally finite complexes using left-modules on rings with identity. Finally, we establish a representation result according to which we can associate a pairing structure with any closable locally finite complex.

LOCALLY FINITE COMPLEXES, MODULES AND GENERALIZED INFORMATION SYSTEMS

Giampiero Chiaselotti
;
Federico Infusino
2022

Abstract

Abstract simplicial complexes (here briefly complexes) are set systems on an arbitrary set which are object of study in many areas of both mathematics and theoretical computer science. Usually, they are investigated over finite sets. However, in general, when we consider an arbitrary set Ω (not necessarily finite) and a complex C on Ω, the most natural property related to finiteness is the following: for any subset X of Ω, if F ∈ C for all finite subsets F of X, then X ∈ C. We call locally finite any complex C having such a property. Bearing in mind some motivations and constructions derived from the analysis of information systems in rough set theory, in this paper we associate with any locally finite complex C a corresponding pre- closure operator σC and, through it, we establish several properties of C. Next, we investigate the main features of the specific sub-class of locally finite complexes C for which σC is a closure operator. We call these complexes closable and exhibit a particular family of closable locally finite complexes using left-modules on rings with identity. Finally, we establish a representation result according to which we can associate a pairing structure with any closable locally finite complex.
Abstract simplicial complexes; closure systems; left-modules; set operators; generalized information systems
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/311981
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