The Generalized de Groot Dual for General Posets
by Martin Kovár, Alena Chernikava
The de Groot dual was originally defined for topological spaces. However, some recent developments in theoretical computer science invoke a growing interest of similar concepts based on slightly different algebraic structures than topologies, in some cases more suitable for applications. In our approach, the dual structure is given by the preframe morphisms (in the sense inspired by the concept of preframe due to B. Banaschewski) from the original poset to the Sierpiński frame. Our application Dual Poset Explorer generates the first three iterated duals of the given poset and displays their Hasse diagrams together with the diagram of the original poset. Posets are expected as embedded into a suitable power set equipped by the inclusion relation as the order, and the preframe morphisms are identified with their the True support (kernel).
The original poset and its three iterated duals:
The legend:
The elements of the first dual:
The elements of the second dual:
The elements of the third dual:
The application Dual Poset Explorer is written in Java powered by Wolfram webMathematica 3.0. The application is hosted at the server of the Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology. Its earlier versions were presented by the authors on several scientific conferences, like Aplimat and International Colloquium in Brno, as an integral part of their research. In case of interest in more theoretical background, see [1], [2] 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"
