generalized toposes with manyvalued logic subobject classifiers
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1 Generalized toposes
1.1 Introduction
Generalized topoi (toposes) with manyvalued algebraic logic subobject classifiers are specified by the associated categories^{} of algebraic logics previously defined as $L{M}_{n}$, that is, noncommutative lattices with $n$ logical values, where $n$ can also be chosen to be any cardinal, including infinity^{}, etc.
1.2 Algebraic category of $L{M}_{n}$ logic algebras
Łukasiewicz logic algebras^{} were constructed by Grigore Moisil in 1941 to define ‘nuances’ in logics, or manyvalued logics, as well as 3state control logic (electronic) circuits^{}. ŁukasiewiczMoisil ($L{M}_{n}$) logic algebras were defined axiomatically in 1970, in ref. [1], as nvalued logic algebra representations and extensions^{} of the Łukasiewcz (3valued) logics; then, the universal properties^{} of categories of $L{M}_{n}$ logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of $L{M}_{n}$logic algebras are under consideration as valid candidates for representations of quantum logics^{}, as well as for modeling nonlinear biodynamics in genetic ‘nets’ or networks ([3]), and in singlecell organisms, or in tumor growth. For a recent review on $n$valued logic algebras, and major published results, the reader is referred to [2].
The category $\mathrm{L}\mathit{}\mathrm{M}$ of ŁukasiewiczMoisil, $n$valued logic algebras ($L\mathit{}{M}_{n}$), and $L\mathit{}{M}_{n}$–lattice^{} morphisms^{}, ${\lambda}_{L{M}_{n}}$, was introduced in 1970 in ref. [1] as an algebraic category^{} tool for $n$valued logic studies. The objects of $\mathcal{L}\mathcal{M}$ are the non–commutative^{} $L{M}_{n}$ lattices and the morphisms of $\mathcal{L}\mathcal{M}$ are the $L{M}_{n}$lattice morphisms as defined next.
Definition 1.1.
A $n$–valued Łukasiewicz–Moisil algebra^{}, ($L{M}_{n}$–algebra) is a structure^{} of the form $(L,\vee ,\wedge ,N,{({\phi}_{i})}_{i\in \{1,\mathrm{\dots},n1\}},0,1)$, subject to the following axioms:

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(L1) $(L,\vee ,\wedge ,N,0,1)$ is a de Morgan algebra, that is, a bounded distributive lattice^{} with a decreasing involution^{} $N$ satisfying the de Morgan property $N(x\vee y)=Nx\wedge Ny$;

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(L2) For each $i\in \{1,\mathrm{\dots},n1\}$, ${\phi}_{i}:L\u27f6L$ is a lattice endomorphism^{};^{$*$}^{$*$}The ${\phi}_{i}$’s are called the Chrysippian endomorphisms^{} of $L$.

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(L3) For each $i\in \{1,\mathrm{\dots},n1\},x\in L$, ${\phi}_{i}(x)\vee N{\phi}_{i}(x)=1$ and ${\phi}_{i}(x)\wedge N{\phi}_{i}(x)=0$;

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(L4) For each $i,j\in \{1,\mathrm{\dots},n1\}$, ${\phi}_{i}\circ {\phi}_{j}={\phi}_{k}$ iff $(i+j)=k$;

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(L5) For each $i,j\in \{1,\mathrm{\dots},n1\}$, $i\u2a7dj$ implies ${\phi}_{i}\u2a7d{\phi}_{j}$;

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(L6) For each $i\in \{1,\mathrm{\dots},n1\}$ and $x\in L$, ${\phi}_{i}(Nx)=N{\phi}_{ni}(x)$.
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Example 1.1.
Let ${L}_{n}=\{0,1/(n1),\mathrm{\dots},(n2)/(n1),1\}$. This set can be naturally endowed with an ${\text{LM}}_{n}$ –algebra structure as follows:

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the bounded lattice^{} operations^{} are those induced by the usual order on rational numbers;

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for each $j\in \{0,\mathrm{\dots},n1\}$, $N(j/(n1))=(nj)/(n1)$;

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for each $i\in \{1,\mathrm{\dots},n1\}$ and $j\in \{0,\mathrm{\dots},n1\}$, ${\phi}_{i}(j/(n1))=0$ if $$ and $=1$ otherwise.
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 ${\text{LM}}_{2}$–algebra, remains consistent^{}.
Example 1.2.
Consider a Boolean algebra $B$.
Let $T(B)=\{({x}_{1},\mathrm{\dots},{x}_{n})\in {B}^{n1}\mid {x}_{1}\u2a7d\mathrm{\dots}\u2a7d{x}_{n1}\}$. On the set $T(B)$, we define an ${\text{LM}}_{n}$algebra structure as follows:

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the lattice operations, as well as $0$ and $1$, are defined component^{}–wise from ${L}_{2}$;

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for each $({x}_{1},\mathrm{\dots},{x}_{n1})\in T(B)$ and $i\in \{1,\mathrm{\dots},n1\}$ one has:
$N({x}_{1},\mathrm{\dots}{x}_{n1})=(\overline{{x}_{n1}},\mathrm{\dots},\overline{{x}_{1}})$ and ${\phi}_{i}({x}_{1},\mathrm{\dots},{x}_{n})=({x}_{i},\mathrm{\dots},{x}_{i}).$
1.3 Generalized logic spaces defined by $L{M}_{n}$ algebraic logics

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Topological semigroup spaces of topological automata

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Topological groupoid^{} spaces of reset automata modules
1.4 Axioms defining a generalized topos

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Consider a subobject logic classifier $O$ defined as an LMalgebraic logic ${L}_{n}$ in the category L of LMlogic algebras, together with logicvalued functors^{} ${F}_{o}:L\to V$, where $V$ is the class of N logic values, with $N$ needing not be finite.

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A triple $(O,\mathbf{L},{F}_{o})$ 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.
More to come…
1.5 Applications of generalized topoi:

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Modern quantum logic (MQL)

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Generalized quantum automata

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Mathematical models of Nstate genetic networks [7]

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Mathematical models of parallel computing networks
References
 1 Georgescu, G. and C. Vraciu. 1970, On the characterization of centered Łukasiewicz algebras., J. Algebra, 16: 486495.
 2 Georgescu, G. 2006, Nvalued Logics and ŁukasiewiczMoisil Algebras, Axiomathes, 16 (12): 123136.
 3 Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Nonlinear Theory. Bulletin of Mathematical Biology, 39: 249258.
 4 Baianu, I.C.: 2004a. ŁukasiewiczTopos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints–Sussex Univ.
 5 Baianu, I.C.: 2004b ŁukasiewiczTopos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT2004059. Health Physics and Radiation Effects (June 29, 2004).
 6 Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and NValued Łukasiewicz Algebras in Relation^{} to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional Algebra^{}, http://en.wikipedia.org/wiki/User:Bci2/Books/InteractomicsAbstract and Preprint of Report in PDF .
 7 Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations^{} of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1–2: 65–122.
 8 Mitchell, Barry. The Theory of Categories. Academic Press: London, 1968.
Title  generalized toposes with manyvalued logic subobject classifiers 
Canonical name  GeneralizedToposesWithManyvaluedLogicSubobjectClassifiers 
Date of creation  20130322 18:13:11 
Last modified on  20130322 18:13:11 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  60 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 58A03 
Classification  msc 18B25 
Classification  msc 03B15 
Classification  msc 03G30 
Classification  msc 03G20 
Classification  msc 03B50 
Synonym  LMnalgebraic nvalued logic 
Synonym  algebraic category of $L{M}_{n}$ logic algebras 
Related topic  NonAbelianStructures 
Related topic  AbelianCategory 
Related topic  AxiomsForAnAbelianCategory 
Related topic  GeneralizedVanKampenTheoremsHigherDimensional 
Related topic  AxiomaticTheoryOfSupercategories 
Related topic  CategoricalOntology 
Related topic  NonCommutingGraphOfAGroup 
Related topic  NonAbelianStructures 
Related topic  QuantumLogicsTopoi 
Related topic  CategoricalAlgebr 
Defines  manyvalued logic subobject classifier 
Defines  algebraic category of LMn logic algebras 
Defines  noncommutative lattice 