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generalized toposes with manyvalued logic subobject classifiers
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 $LM_{n}$, 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 $LM_{n}$ 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 ($LM_{n}$) 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 $LM_{n}$ logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of $LM_{n}$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 $\mathcal{LM}$ of ŁukasiewiczMoisil, $n$valued logic algebras ($LM_{n}$), and $LM_{n}$–lattice morphisms, $\lambda_{{LM_{n}}}$, was introduced in 1970 in ref. [1] as an algebraic category tool for $n$valued logic studies. The objects of $\mathcal{LM}$ are the non–commutative $LM_{n}$ lattices and the morphisms of $\mathcal{LM}$ are the $LM_{n}$lattice morphisms as defined next.
Definition 1.1.
A $n$–valued Łukasiewicz–Moisil algebra, ($LM_{{n}}$–algebra) is a structure of the form $(L,\vee,\wedge,N,(\varphi_{{i}})_{{i\in\{1,\ldots,n1\}}},0,1)$, subject to the following axioms:

(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 $N({x\vee y})=Nx\wedge Ny$;

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

(L3) For each $i\in\{1,\ldots,n1\},x\in L$, $\varphi_{{i}}(x)\vee N{\varphi_{{i}}(x)}=1$ and $\varphi_{{i}}(x)\wedge N{\varphi_{{i}}(x)}=0$;

(L4) For each $i,j\in\{1,\ldots,n1\}$, $\varphi_{{i}}\circ\varphi_{{j}}=\varphi_{{k}}$ iff $(i+j)=k$;

(L5) For each $i,j\in\{1,\ldots,n1\}$, $i\leqslant j$ implies $\varphi_{{i}}\leqslant\varphi_{{j}}$;

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

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

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

for each $i\in\{1,\ldots,n1\}$ and $j\in\{0,\ldots,n1\}$, $\varphi_{{i}}(j/(n1))=0$ if $j<i$ and $=1$ otherwise.
Note that, for $n=2$, $L_{n}=\{0,1\}$, and there is only one Chrysippian endomorphism of $L_{n}$ is $\varphi_{1}$, 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 $\varphi_{1}$). 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.
Example 1.2.
Consider a Boolean algebra $B$.
Let $T(B)=\{(x_{1},\ldots,x_{n})\in B^{{n1}}\mid x_{1}\leqslant\ldots\leqslant x_{% {n1}}\}$. On the set $T(B)$, we define an $\mbox{LM}_{n}$algebra structure as follows:

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

for each $(x_{1},\ldots,x_{{n1}})\in T(B)$ and $i\in\{1,\ldots,n1\}$ one has:
$N(x_{1},\ldots x_{{n1}})=(\overline{x_{{n1}}},\ldots,\overline{x_{1}})$ and $\varphi_{{i}}(x_{1},\ldots,x_{n})=(x_{i},\ldots,x_{i}).$
1.3 Generalized logic spaces defined by $LM_{n}$ algebraic logics

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

A triple $(O,{\bf L},F_{o})$ 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:

Modern quantum logic (MQL)

Generalized quantum automata

Mathematical models of Nstate genetic networks [7]

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, Abstract 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.
Mathematics Subject Classification
58A03 no label found18B25 no label found03B15 no label found03G30 no label found03G20 no label found03B50 no label found Forums
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