# 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 categories of algebraic logics previously defined as ${\displaystyle LM_{n}}$ , that is, non-commutative lattices with ${\displaystyle n}$  logical values, where ${\displaystyle n}$  can also be chosen to be any cardinal, including infinity, etc.

### Algebraic category of ${\displaystyle LM_{n}}$  logic algebras

\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 (${\displaystyle LM_{n}}$ ) 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 ${\displaystyle LM_{n}}$  -logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of ${\displaystyle LM_{n}}$ -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 ${\displaystyle n}$ -valued logic algebras, and major published results, the reader is referred to [2].

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

\rm

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

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

\begin{exe}\rm Let ${\displaystyle L_{n}=\{0,1/(n-1),\ldots ,(n-2)/(n-1),1\}}$ . This set can be naturally endowed with an ${\displaystyle {\mbox{LM}}_{n}}$  --algebra structure as follows:

• the bounded lattice operations are those induced by the usual order on rational numbers;
• for each ${\displaystyle j\in \{0,\ldots ,n-1\}}$ , ${\displaystyle N(j/(n-1))=(n-j)/(n-1)}$ ;
• for each ${\displaystyle i\in \{1,\ldots ,n-1\}}$  and ${\displaystyle j\in \{0,\ldots ,n-1\}}$ , $\displaystyle \phii(j/(n-1))=0$ if ${\displaystyle j  and ${\displaystyle =1}$  otherwise.

\end{exe} Note that, for ${\displaystyle n=2}$ , ${\displaystyle L_{n}=\{0,1\}}$ , and there is only one Chrysippian endomorphism of ${\displaystyle L_{n}}$  is ${\displaystyle \phi _{1}}$ , which is necessarily restricted by the determination principle to a bijection, thus making ${\displaystyle L_{n}}$  a Boolean algebra (if we were also to disregard the redundant bijection ${\displaystyle \phi _{1}}$ ). Hence, the overloaded' notation ${\displaystyle L_{2}}$ , which is used for both the classical Boolean algebra and the two--element ${\displaystyle {\mbox{LM}}_{2}}$ --algebra, remains consistent. \begin{exe}\rm Consider a Boolean algebra $\displaystyle (B,\v,\w,{}^-,0,1)$ . Let ${\displaystyle T(B)=\{(x_{1},\ldots ,x_{n})\in B^{n-1}\mid x_{1}\leq \ldots \leq x_{n-1}\}$.Ontheset}$ T(B)${\displaystyle ,wedefinean}$ \mbox{LM}_n$-algebra structure as follows:

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

\end{exe}

## References

1. Georgescu, G. and C. Vraciu. 1970, On the characterization of centered \L{}ukasiewicz algebras., J. Algebra , 16 : 486-495.
2. Georgescu, G. 2006, N-valued Logics and \L ukasiewicz-Moisil Algebras, Axiomathes , 16 (1-2): 123-136.
3. 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. 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 .