# 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

- 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 SourcesEdit

^{[1]}^{[2]}^{[3]}^{[6]}^{[7]}^{[8]}^{[5]}

## ReferencesEdit

- ↑
^{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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tag; no text was provided for refs named`GG-CV70,GG2k6`

- ↑
^{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 .