PlanetPhysics/Generalized Topoi With LMn Algebraic Logic Classifiers
Generalized toposes
editIntroduction
editGeneralized 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 algebras
edit\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 logics
edit- topological semigroup spaces of topological automata topological groupoid spaces of reset automata modules
Applications of generalized topoi:
edit- Modern quantum logic (MQL)
- Generalized quantum automata
- Mathematical models of N-state genetic networks [5]
- Mathematical models of parallel computing networks
All Sources
editReferences
edit- ↑ 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.
- ↑ 2.0 2.1 2.2 Georgescu, G. 2006, N-valued Logics and \L ukasiewicz-Moisil Algebras, Axiomathes , 16 (1-2): 123-136.
- ↑ 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.
- ↑ Cite error: Invalid
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- ↑ 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.
- ↑ Baianu, I.C.: 2004a. \L{}ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints--Sussex Univ.
- ↑ 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).
- ↑ 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 .