GeneralizedToposesWithManyValuedLogicSubobjectClassifiers 2009-05-01 00:22:17 2010-12-22 14:37:12 Topic generalized toposes many-valued logic subobject classifiers generalized toposes \subsection{Introduction} \emph{Generalized topoi (toposes) with many-valued algebraic logic subobject classifiers} are specified by the associated categories of algebraic logics previously defined as $LM_n$, that is, {\em non-commutative} lattices with $n$ logical values, where $n$ can also be chosen to be any cardinal, including infinity, etc. \subsection{Algebraic category of $LM_n$ logic algebras} \L{}ukasiewicz \emph{logic algebras} were constructed by Grigore Moisil in 1941 to define `nuances' in logics, or many-valued logics, as well as 3-state control logic (electronic) circuits. \L{}ukasiewicz-Moisil ($LM_n$) logic algebras were defined axiomatically in 1970, in ref. \cite{GG-CV70}, as n-valued logic algebra representations and extensions of the \L ukasiewcz (3-valued) logics; then, the universal properties of categories of $LM_n$ -logic algebras were also investigated and reported in a series of recent publications (\cite{GG2k6} and references cited therein). Recently, several modifications of {\em $LM_n$-logic algebras} are under consideration as valid candidates for representations of {\em quantum logics}, as well as for modeling non-linear biodynamics in genetic `nets' or networks (\cite{ICB77}), and in single-cell organisms, or in tumor growth. For a recent review on $n$-valued logic algebras, and major published results, the reader is referred to \cite{GG2k6}. \subsection{Generalized logic spaces defined by $LM_n$ algebraic logics} \begin{itemize} \item Topological semigroup spaces of topological automata \item Topological groupoid spaces of reset automata modules \end{itemize} \subsection{Axioms defining generalized topoi} \begin{itemize} \item Consider a subobject logic classifier $O$ defined as an LM-algebraic logic $L_n$ in the category $L$ of LM-logic algebras, together with logic-valued functors $Fo: L \to V$, where $V$ is the class of N logic values, with $N$ needing not be finite. \item A triple $(O,L,Fo)$ defines a generalized topos, $\tau$, if the above axioms defining $O$ are satisfied, and if the functor $Fo$ is an univalued functor in the sense of Mitchell. \end{itemize} {\bf More to come...} \subsection{Applications of generalized topoi:} \begin{itemize} \item Modern quantum logic (MQL) \item Generalized quantum automata \item Mathematical models of N-state genetic networks \cite{BBGG1} \item Mathematical models of parallel computing networks \end{itemize} \subsection{Applications of generalized topoi:} \begin{itemize} \item XY \item YZ \end{itemize} \subsection{Generalized logic `spaces' defined by LMn.} \begin{itemize} \item XY \item YZ \end{itemize} \begin{thebibliography}{9} \bibitem{GG-CV70} Georgescu, G. and C. Vraciu. 1970, On the characterization of centered \L{}ukasiewicz algebras., {\em J. Algebra}, \textbf{16}: 486-495. \bibitem{GG2k6} Georgescu, G. 2006, N-valued Logics and \L ukasiewicz-Moisil Algebras, \emph{Axiomathes}, \textbf{16} (1-2): 123-136. \bibitem{ICB77} Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. \emph{Bulletin of Mathematical Biology}, \textbf{39}: 249-258. \bibitem{ICB2004a} Baianu, I.C.: 2004a. \L{}ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints--Sussex Univ. \bibitem{ICB04b} Baianu, I.C.: 2004b \L{}ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. \textit{Health Physics and Radiation Effects} (June 29, 2004). \bibitem{Bgg2} Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, \textbf{(M,R)}--Systems and Their Higher Dimensional Algebra, \PMlinkexternal{Abstract and Preprint of Report in PDF}{http://www.ag.uiuc.edu/fs401/QAuto.pdf} . \bibitem{BBGG1} Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L{}ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., \emph{Axiomathes}, \textbf{16} Nos. 1--2: 65--122. \end{thebibliography}