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%%% Primary Title: generalized topoi with LMn algebraic logic classifiers
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\begin{document}
\section{Generalized toposes}
\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}.
The \emph{category $\mathcal{LM}$ of \L{}ukasiewicz-Moisil, $n$-valued logic algebras ($LM_n$), and $LM_n$--lattice morphisms}, $\lambda_{LM_n}$, was introduced in 1970 in ref. \cite{GG-CV70} as an \htmladdnormallink{algebraic category}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} tool for $n$-valued logic studies. The \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $\mathcal{LM}$ are the \emph{non--commutative} $LM_n$ lattices and the \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $\mathcal{LM}$ are the $LM_n$-lattice morphisms as defined next.
\begin{definition}\rm
A {\it $n$--valued \L ukasiewicz--Moisil algebra}, ({\it $LM_{n}$--algebra}) is a structure of the form
$(L,\vee,\wedge,N,(\phii)_{i\in\{1,\ldots,n-1\}},0,1)$, subject to the following axioms:
\begin{itemize}
\item (L1) $(L,\vee,\wedge,N,0,1)$ is a {\it de Morgan algebra}, that is, a bounded distributive lattice with a decreasing involution $N$ satisfying the de Morgan property $N({x\vee y})=Nx\wedge Ny$;
\item (L2) For each $i\in\{1,\ldots,n-1\}$, $\phii:L\lra L$ is a lattice endomorphism;\footnote{ The $\phii$'s are called the {\em Chrysippian endomorphisms} of $L$.}
\item (L3) For each $i\in\{1,\ldots,n-1\},x\in L$, $\phii(x)\vee N{\phii(x)}=1$ and
$\phii(x)\wedge N{\phii(x)}=0$;
\item (L4) For each $i,j\in\{1,\ldots,n-1\}$, $\phii\circ\phi_{j}=\phi_{k}$ iff $(i+j)= k$;
\item (L5) For each $i,j\in\{1,\ldots,n-1\}$, $i\leq j$ implies $\phii\leq\phi_{j}$;
\item (L6) For each $i\in\{1,\ldots,n-1\}$ and $x\in L$, $\phii(N x)=N\phi_{n-i}(x)$.
\item (L7) Moisil's `determination principle':
$$\left[\orc i\in\{1,\ldots,n-1\},\;\phii(x)=\phii(y)\right] \; implies \; [x = y] \;$$
\cite{GG-CV70,GG2k6}.
\end{itemize}
\end{definition}
\begin{exe}\rm
Let $L_n=\{0,1/(n-1),\ldots,(n-2)/(n-1),1\}$. This set can be naturally endowed with an $\mbox{LM}_n$
--algebra structure as follows:
\begin{itemize}
\item the bounded lattice \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} are those induced by the usual order on rational numbers;
\item for each $j\in\{0,\ldots,n-1\}$, $N(j/(n-1))=(n-j)/(n-1)$;
\item for each $i\in\{1,\ldots,n-1\}$ and $j\in\{0,\ldots,n-1\}$,
$\phii(j/(n-1))=0$ if $j<i$ and $=1$ otherwise.
\end{itemize}
\end{exe}
Note that, for $n=2$, $L_n=\{0,1\}$, and there is only one Chrysippian endomorphism of $L_n$ is $\phi_1$, which
is necessarily restricted by the determination principle to a bijection, thus making $L_n$ a Boolean algebra (if
we were also to disregard the redundant bijection $\phi_1$). Hence, the `overloaded' notation $L_2$, which is
used for both the classical Boolean algebra and the two--element $\mbox{LM}_2$--algebra, remains consistent.
\begin{exe}\rm
Consider a Boolean algebra $(B,\v,\w,{}^-,0,1)$. Let $T(B)=\{(x_1,\ldots,x_n)\in B^{n-1}\mid x_1\leq\ldots\leq
x_{n-1}\}$. On the set $T(B)$, we define an $\mbox{LM}_n$-algebra structure as follows:
\begin{itemize}
\item the lattice operations, as well as $0$ and $1$, are defined component--wise from $\Ld$;
\item for each $(x_1,\ldots,x_{n-1})\in T(B)$ and $i\in\{1,\ldots,n-1\}$ one has:\\
$N(x_1,\ldots x_{n-1})=(\ov{x_{n-1}},\ldots,\ov{x_1})$ and $\phii(x_1,\ldots,x_n)=(x_i,\ldots,x_i) .$
\end{itemize}
\end{exe}
\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{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}
\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://en.wikipedia.org/wiki/User:Bci2/Books/Interactomics} .
\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}
