View source for PlanetPhysics/Generalized Topoi With LMn Algebraic Logic Classifiers

==Generalized toposes==
===Introduction===

''Generalized topoi (toposes) with many-valued \htmladdnormallink{algebraic'' {http://planetphysics.us/encyclopedia/CoIntersections.html} logic subobject classifiers} are specified by the associated [[../Cod/|categories]] of algebraic logics previously defined as <math>LM_n</math>, that is, ''[[../AbelianCategory3/|non-commutative]]'' lattices with <math>n</math> logical values, where <math>n</math> can also be chosen to be any cardinal, including infinity, etc.

===Algebraic category of <math>LM_n</math> logic algebras===

\L{}ukasiewicz ''logic algebras''  were constructed by Grigore Moisil in 1941 to define `nuances' in logics, or [[../LM_nLogicAlgebra/|many-valued logics]], as well as 3-state control logic (electronic) circuits. \L{}ukasiewicz-Moisil (<math>LM_n</math>) logic algebras were defined axiomatically in 1970, in ref. <ref name='GG-CV70'/>, as [[../LM_nLogicAlgebra/|N-valued logic algebra]] [[../CategoricalGroupRepresentation/|representations]] and extensions of the \L ukasiewcz (3-valued) logics; then, the universal properties of categories of <math>LM_n</math> -logic algebras were also investigated and reported in a series of recent publications (<ref name='GG2k6'/> and references cited therein). Recently, several modifications of ''<math>LM_n</math>-logic algebras''  are under consideration as valid candidates for representations of ''[[../LQG2/|quantum logics]]'', as well as for modeling non-linear biodynamics in genetic `nets' or networks (<ref name='ICB77'/>), and in single-cell organisms, or in tumor growth. For a recent review on <math>n</math>-valued logic algebras, and major published results, the reader is referred to <ref name='GG2k6'/>.

The category <math>\mathcal{LM{{'}}{{'}} </math> of \L{}ukasiewicz-Moisil, <math>n</math>-valued logic algebras (<math>LM_n</math>), and <math>LM_n</math>--lattice morphisms}, <math>\lambda_{LM_n}</math>, was introduced in 1970 in ref. <ref name='GG-CV70'/> as an [[../CategoryOfLogicAlgebras/|algebraic category]] tool for <math>n</math>-valued logic studies. The [[../TrivialGroupoid/|objects]] of <math>\mathcal{LM}</math> are the ''non--commutative''  <math>LM_n</math> lattices and the [[../TrivialGroupoid/|morphisms]] of <math>\mathcal{LM}</math> are the <math>LM_n</math>-lattice morphisms as defined next.

\rm

A {\it <math>n</math>--valued \L ukasiewicz--Moisil algebra}, ({\it <math>LM_{n}</math>--algebra}) is a structure of the form
<math>(L,\vee,\wedge,N,(\phii)_{i\in\{1,\ldots,n-1\}},0,1)</math>, subject to the following axioms:

*(L1) <math>(L,\vee,\wedge,N,0,1)</math> is a {\it de Morgan algebra}, that is, a bounded distributive lattice with a decreasing involution <math>N</math> satisfying the de Morgan property <math>N({x\vee y})=Nx\wedge Ny</math>;
*(L2) For each <math>i\in\{1,\ldots,n-1\}</math>, <math>\phii:L\lra L</math> is a lattice endomorphism;\footnote{ The <math>\phii</math>'s are called the ''Chrysippian endomorphisms''  of <math>L</math>.}
*(L3) For each <math>i\in\{1,\ldots,n-1\},x\in L</math>, <math>\phii(x)\vee N{\phii(x)}=1</math> and <math>\phii(x)\wedge N{\phii(x)}=0</math>;
*(L4) For each <math>i,j\in\{1,\ldots,n-1\}</math>, <math>\phii\circ\phi_{j}=\phi_{k}</math> iff <math>(i+j)= k</math>;
*(L5) For each <math>i,j\in\{1,\ldots,n-1\}</math>, <math>i\leq j</math> implies <math>\phii\leq\phi_{j}</math>;
*(L6) For each <math>i\in\{1,\ldots,n-1\}</math> and <math>x\in L</math>, <math>\phii(N x)=N\phi_{n-i}(x)</math>.
*(L7) Moisil's `determination principle': <math>\left[\orc i\in\{1,\ldots,n-1\},\;\phii(x)=\phii(y)\right] \; implies \; [x = y] \;</math> <ref name='GG-CV70,GG2k6'/>.

\begin{exe}\rm
Let <math>L_n=\{0,1/(n-1),\ldots,(n-2)/(n-1),1\}</math>. This set can be naturally endowed with an <math>\mbox{LM}_n</math>
--algebra structure as follows:

*the bounded lattice [[../Cod/|operations]] are those induced by the usual order on rational numbers;
*for each <math>j\in\{0,\ldots,n-1\}</math>, <math>N(j/(n-1))=(n-j)/(n-1)</math>;
*for each <math>i\in\{1,\ldots,n-1\}</math> and <math>j\in\{0,\ldots,n-1\}</math>, <math>\phii(j/(n-1))=0</math> if <math>j<i</math> and <math>=1</math> otherwise.
\end{exe}
Note that, for <math>n=2</math>, <math>L_n=\{0,1\}</math>, and there is only one Chrysippian endomorphism of <math>L_n</math> is <math>\phi_1</math>, which
is necessarily restricted by the determination principle to a bijection, thus making <math>L_n</math> a Boolean algebra (if
we were also to disregard the redundant bijection <math>\phi_1</math>). Hence, the `overloaded' notation <math>L_2</math>, which is
used for both the classical Boolean algebra and the two--element <math>\mbox{LM}_2</math>--algebra, remains consistent.
\begin{exe}\rm
Consider a Boolean algebra <math>(B,\v,\w,{}^-,0,1)</math>. Let <math>T(B)=\{(x_1,\ldots,x_n)\in B^{n-1}\mid x_1\leq\ldots\leq
x_{n-1}\}<math>. On the set </math>T(B)<math>, we define an </math>\mbox{LM}_n</math>-algebra structure as follows:

*the lattice operations, as well as <math>0</math> and <math>1</math>, are defined component--wise from <math>\Ld</math>;
*for each <math>(x_1,\ldots,x_{n-1})\in T(B)</math> and <math>i\in\{1,\ldots,n-1\}</math> one has:\\ <math>N(x_1,\ldots x_{n-1})=(\ov{x_{n-1}},\ldots,\ov{x_1})</math> and <math>\phii(x_1,\ldots,x_n)=(x_i,\ldots,x_i) .</math>
\end{exe}

===Generalized logic spaces defined by <math>LM_n</math> algebraic logics===

*[[../CoIntersections/|topological]] [[../TrivialGroupoid/|semigroup]] spaces of topological automata [[../GroupoidHomomorphism2/|topological groupoid]] spaces of reset automata [[../RModule/|modules]]

===Applications of generalized topoi:===

*Modern quantum logic (MQL)
*Generalized [[../QuantumComputers/|quantum automata]]
*Mathematical models of N-state [[../GeneNetDigraph/|genetic networks]] <ref name='BBGG1'/>
*Mathematical models of parallel computing networks

==All Sources==
<ref name='GG-CV70'/>
<ref name='GG2k6'/>
<ref name='ICB77'/>
<ref name='ICB2004a'/>
<ref name='ICB04b'/>
<ref name='Bgg2'/>
<ref name='BBGG1'/>

==References==
<references>

</ref><ref name='GG-CV70'>
Georgescu, G. and C. Vraciu. 1970, On the characterization of centered \L{}ukasiewicz
algebras., ''J. Algebra'' , '''16''' : 486-495.

</ref><ref name='GG2k6'>
Georgescu, G. 2006, N-valued Logics and \L ukasiewicz-Moisil Algebras, ''Axiomathes'' , '''16'''  (1-2): 123-136.

</ref><ref name='ICB77'>
Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. ''Bulletin of Mathematical Biology'' , '''39''' : 249-258.

</ref><ref name='ICB2004a'>
Baianu, I.C.: 2004a. \L{}ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints--Sussex Univ.

</ref><ref name='ICB04b'>
Baianu, I.C.: 2004b \L{}ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. ''Health Physics and Radiation Effects''  (June 29, 2004).

</ref><ref name='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, '''(M,R)''' --Systems and Their Higher Dimensional Algebra,
[[Wikipedia:User:Bci2/Books/Interactomics|Abstract and Preprint of Report in PDF]] .

</ref><ref name='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., ''Axiomathes'' , '''16'''  Nos. 1--2: 65--122.

</ref></references>

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