# generalized toposes with many-valued logic subobject classifiers

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## 1 Generalized toposes

### 1.1 Introduction

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, non-commutative 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 many-valued logics, as well as 3-state control logic (electronic) circuits. Łukasiewicz-Moisil ($LM_{n}$) logic algebras were defined axiomatically in 1970, in ref. [1], as n-valued logic algebra representations and extensions of the Łukasiewcz (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 ([2] and references cited therein). Recently, several modifications of $LM_{n}$-logic algebras are under consideration as valid candidates for representations of , as well as for modeling non-linear biodynamics in genetic ‘nets’ or networks ([3]), 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 [2].

The category $\mathcal{LM}$ of Łukasiewicz-Moisil, $n$-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 $(L,\vee,\wedge,N,(\varphi_{i})_{i\in\{1,\ldots,n-1\}},0,1)$, 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 $i\in\{1,\ldots,n-1\}$, $\varphi_{i}:L{\longrightarrow}L$ is a lattice endomorphism;$*$$*$The $\varphi_{i}$’s are called the Chrysippian endomorphisms of $L$.

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

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

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

• (L6) For each $i\in\{1,\ldots,n-1\}$ and $x\in L$, $\varphi_{i}(Nx)=N\varphi_{n-i}(x)$.

• (L7) Moisil’s ‘determination principle’:

 $\left[\forall i\in\{1,\ldots,n-1\},\;\varphi_{i}(x)=\varphi_{i}(y)\right]\;% implies\;[x=y]\;$

[1, 2].

###### Example 1.1.

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:

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

• for each $j\in\{0,\ldots,n-1\}$, $N(j/(n-1))=(n-j)/(n-1)$;

• for each $i\in\{1,\ldots,n-1\}$ and $j\in\{0,\ldots,n-1\}$, $\varphi_{i}(j/(n-1))=0$ if $j 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 $T(B)=\{(x_{1},\ldots,x_{n})\in B^{n-1}\mid x_{1}\leqslant\ldots\leqslant x_{n-% 1}\}$. 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 $(x_{1},\ldots,x_{n-1})\in T(B)$ and $i\in\{1,\ldots,n-1\}$ one has:
$N(x_{1},\ldots x_{n-1})=(\overline{x_{n-1}},\ldots,\overline{x_{1}})$ and $\varphi_{i}(x_{1},\ldots,x_{n})=(x_{i},\ldots,x_{i}).$

### 1.4 Axioms defining a generalized topos

• 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 $F_{o}:L\to V$, where $V$ is the class of N logic values, with $N$ needing not be finite.

• 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 N-state 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: 486-495.
• 2 Georgescu, G. 2006, N-valued Logics and Łukasiewicz-Moisil Algebras, Axiomathes, 16 (1-2): 123-136.
• 3 Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology, 39: 249-258.
• 4 Baianu, I.C.: 2004a. Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints–Sussex Univ.
• 5 Baianu, I.C.: 2004b Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).
• 6 Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued 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 many-valued logic subobject classifiers Canonical name GeneralizedToposesWithManyvaluedLogicSubobjectClassifiers Date of creation 2013-03-22 18:13:11 Last modified on 2013-03-22 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 LMn-algebraic n-valued logic Synonym algebraic category of $LM_{n}$ 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 many-valued logic subobject classifier Defines algebraic category of LMn logic algebras Defines noncommutative lattice