Talk:PlanetPhysics/Generalized Toposes With Many Valued Logic Subobject Classifiers

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: generalized toposes with many-valued logic subobject classifiers %%% Primary Category Code: 00. %%% Filename: GeneralizedToposesWithManyValuedLogicSubobjectClassifiers.tex %%% Version: 3 %%% Owner: bci1 %%% Author(s): bci1 %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

\begin{document}

\subsection{Introduction}

\emph{Generalized topoi (toposes) with many-valued \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} logic subobject classifiers} are specified by the associated \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} of algebraic logics previously defined as $LM_n$, that is, {\em \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}} 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 \htmladdnormallink{many-valued logics}{http://planetphysics.us/encyclopedia/LM_nLogicAlgebra.html}, 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 \htmladdnormallink{N-valued logic algebra}{http://planetphysics.us/encyclopedia/LM_nLogicAlgebra.html} \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} 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 \htmladdnormallink{quantum logics}{http://planetphysics.us/encyclopedia/LQG2.html}}, 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 \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{semigroup}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} spaces of topological automata \item \htmladdnormallink{topological groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} spaces of reset automata \htmladdnormallink{modules}{http://planetphysics.us/encyclopedia/RModule.html} \end{itemize}

\subsection{Axioms defining generalized topoi} \begin{itemize} \item Consider a subobject logic classifier $\Omega$ defined as an LM-algebraic logic $L_n$ in the category ${\bf L}$ of LM-logic algebras, together with logic-valued \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $F_{\omega}: {\bf L} \to V$, where $V$ is the class of N logic values, with $N$ needing not be finite. \item A triple $(\Omega,L,F_{\omega})$ defines a generalized \htmladdnormallink{topos}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html}, $\tau$, if the above axioms defining $\Omega$ are satisfied, and if the functor $F_{\omega}$ 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 \htmladdnormallink{quantum automata}{http://planetphysics.us/encyclopedia/QuantumComputers.html} \item Mathematical models of N-state \htmladdnormallink{genetic networks}{http://planetphysics.us/encyclopedia/GeneNetDigraph.html} \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, \htmladdnormallink{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}

\end{document}