Framework Isomorphisms

  by   Martin Kovár, Alena Chernikava, Štěpán Křehlík

Abstract and usage:

Framework is an algebraic structure whose purpose is capturing the topological or topology-like structure of a system from its externally observed properties. Formally, framework is a formal context whose incidence relation is the membership relation. The objects are called places and the attributes are collections of places connected by a possible presence of some physical object, e.g., a particle. The set of attributes is called a framology. The structure may be used, for example, for investigation of topological properties of certain causal structures, motivated by quantum gravity, interactions of particles, Feynman diagrams, information systems and databases or solutions of certain differential equations and their global properties. They also may be used for representation of formal contexts and for natural topologization of structures of formal concept analysis. Due to possible symmetries in frameworks, there may exist more than one isomorphisms between two frameworks. Thus, the isomorphisms may be used for study the symmetries of frameworks, which may be applied in topological modeling. For more exact definitions and some details, the reader is referred to [1]. Our presented application Framework Isomorphism Generator generates the set of all isomorphisms between two given frameworks and allows selecting one of them for further processing or investigation.

The first framework data:

The set of places:

The framology:

The second framework data:

The set of places:

The framology:

  Selected isomorphism:  


The first framework:

The second framework:

The list of framework isomorphisms:

Number of framework isomorphisms:

The selected isomorphism:


The application Framework Isomorphism Generator is written in Java powered by Wolfram webMathematica 3.1. The application is hosted at the server of the Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology. The results representing its theoretical background were presented by the authors on several scientific conferences as an integral part of their research. In case of interest in more detail, see [1] or contact the authors. For research and scientific activities the software is available free of charge. In all other cases, please contact RNDr. M. Novák, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, phone: +420 5 4114 3135. Acknowledgement: FEKT-S-11-2/921 "Vlastnosti řešení funkcionálních diferenciálních a diferenčních rovnic"

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