PlanetPhysics/Generalized Topoi With LMn Algebraic Logic Classifiers

Generalized toposesEdit

IntroductionEdit

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

Algebraic category of   logic algebrasEdit

\L{}ukasiewicz 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. \L{}ukasiewicz-Moisil ( ) logic algebras were defined axiomatically in 1970, in ref. [1], as N-valued logic algebra representations and extensions of the \L ukasiewcz (3-valued) logics; then, the universal properties of categories of   -logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of  -logic algebras are under consideration as valid candidates for representations of quantum logics, 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  -valued logic algebras, and major published results, the reader is referred to [2].

The category Failed to parse (syntax error): {\displaystyle \mathcal{LM{{'}}{{'}} } of \L{}ukasiewicz-Moisil,  -valued logic algebras ( ), and  --lattice morphisms},  , was introduced in 1970 in ref. [1] as an algebraic category tool for  -valued logic studies. The objects of   are the non--commutative   lattices and the morphisms of   are the  -lattice morphisms as defined next.

\rm

A {\it  --valued \L ukasiewicz--Moisil algebra}, ({\it  --algebra}) is a structure of the form Failed to parse (unknown function "\phii"): {\displaystyle (L,\vee,\wedge,N,(\phii)_{i\in\{1,\ldots,n-1\}},0,1)} , subject to the following axioms:

  • (L1)   is a {\it de Morgan algebra}, that is, a bounded distributive lattice with a decreasing involution   satisfying the de Morgan property  ;
  • (L2) For each  , Failed to parse (unknown function "\phii"): {\displaystyle \phii:L\lra L} is a lattice endomorphism;\footnote{ The Failed to parse (unknown function "\phii"): {\displaystyle \phii} 's are called the Chrysippian endomorphisms of  .}
  • (L3) For each  , Failed to parse (unknown function "\phii"): {\displaystyle \phii(x)\vee N{\phii(x)}=1} and Failed to parse (unknown function "\phii"): {\displaystyle \phii(x)\wedge N{\phii(x)}=0} ;
  • (L4) For each  , Failed to parse (unknown function "\phii"): {\displaystyle \phii\circ\phi_{j}=\phi_{k}} iff  ;
  • (L5) For each  ,   implies Failed to parse (unknown function "\phii"): {\displaystyle \phii\leq\phi_{j}} ;
  • (L6) For each   and  , Failed to parse (unknown function "\phii"): {\displaystyle \phii(N x)=N\phi_{n-i}(x)} .
  • (L7) Moisil's `determination principle': Failed to parse (unknown function "\orc"): {\displaystyle \left[\orc i\in\{1,\ldots,n-1\},\;\phii(x)=\phii(y)\right] \; implies \; [x = y] \;} [4].

\begin{exe}\rm Let  . This set can be naturally endowed with an   --algebra structure as follows:

  • the bounded lattice operations are those induced by the usual order on rational numbers;
  • for each  ,  ;
  • for each   and  , Failed to parse (unknown function "\phii"): {\displaystyle \phii(j/(n-1))=0} if   and   otherwise.

\end{exe} Note that, for  ,  , and there is only one Chrysippian endomorphism of   is  , which is necessarily restricted by the determination principle to a bijection, thus making   a Boolean algebra (if we were also to disregard the redundant bijection  ). Hence, the `overloaded' notation  , which is used for both the classical Boolean algebra and the two--element  --algebra, remains consistent. \begin{exe}\rm Consider a Boolean algebra Failed to parse (unknown function "\v"): {\displaystyle (B,\v,\w,{}^-,0,1)} . Let  T(B) \mbox{LM}_n</math>-algebra structure as follows:

  • the lattice operations, as well as   and  , are defined component--wise from Failed to parse (unknown function "\Ld"): {\displaystyle \Ld} ;
  • for each   and   one has:\\ Failed to parse (unknown function "\ov"): {\displaystyle N(x_1,\ldots x_{n-1})=(\ov{x_{n-1}},\ldots,\ov{x_1})} and Failed to parse (unknown function "\phii"): {\displaystyle \phii(x_1,\ldots,x_n)=(x_i,\ldots,x_i) .}

\end{exe}

Generalized logic spaces defined by   algebraic logicsEdit

Applications of generalized topoi:Edit

All SourcesEdit

[1][2][3][6][7][8][5]

ReferencesEdit

  1. 1.0 1.1 1.2 Georgescu, G. and C. Vraciu. 1970, On the characterization of centered \L{}ukasiewicz algebras., J. Algebra , 16 : 486-495.
  2. 2.0 2.1 2.2 Georgescu, G. 2006, N-valued Logics and \L ukasiewicz-Moisil Algebras, Axiomathes , 16 (1-2): 123-136.
  3. 3.0 3.1 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.
  4. Cite error: Invalid <ref> tag; no text was provided for refs named GG-CV70,GG2k6
  5. 5.0 5.1 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.
  6. Baianu, I.C.: 2004a. \L{}ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints--Sussex Univ.
  7. 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).
  8. 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, Abstract and Preprint of Report in PDF .