A representation theorem for co-diagonalizable algebras

• Autorzy: Buszkowski W.
• Języki publikacji: EN

Abstrakty

EN

Tadeusz Prucnal [5] published a proof of the following representation theorem: for every atomic co-diagonalizable algebra D there exists an embedding h form D into the field of all subsets of a topological space X such that, for all a ? D, h (?(a)) is the derivative of the set h (a). He presented this results on a conference in Poland in 1983 and left open the question of it could be generalized to all co-diagonizable algebras. It was my observation that some ideas of measure theory (extending a measure to a complete measure) could be applied to define an embedding of any co-diagonalizable algebra into an atomic co-diagonalizable algebra. Consequently, Prucnal's representation theorem holds true for arbitrary co-diagonalizable algebra, witch has been published in our common paper [3]. Two years alter, I've published a short note [2] examining a more general situation of embeddability of modal structures into atomic modal structures. This paper surveys these results with modified proofs and additional comments.

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2004
• Tom: Nr 38
• Strony: 13--22

Twórcy

autor

Buszkowski W.

• Faculty of Mathematics and Computerv Science, Adam Mickiewicz University,

Bibliografia

• [1] G. Boolos, The Unprovability of Consistency, Cambridge University Press, Cambridge, 1979.
• [2] W. Buszkowski, Embedding Boolean structures into atomic Boolean structures, Zeitschrift, für mathematische Logik und Grundlagen der Mathematik 32 (1986), 227-228.
• [3] W. Buszkowski and T. Prucnal, Topological representation of co-diagonalizable algebras, in: G. Wechsung (ed.), Frege Conference’ 1984, Akademie-Verlag, Berlin, 1984, 63-65.
• [4] R. Magari, Representation and duality theory for diagonalizable algebras, Studio. Logica 34 (1975), 305-313.
• [5] T. Prucnal, Topological representation of atomic co-diagonalizable algebras, Bulletin of The Section of Logic 12 (1983), 71-72.
• [6] R.M. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics 25 (1976), 287-304.
• [7] C. Smoryński, Modal logic and self-reference, in: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. 2, Reidel, Dordrecht, 1984, 441-495.