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 $LM_{n}$, that is, non-commutative lattices with $n$ logical values, where $n$ can also be chosen to be any cardinal, including infinity, etc.

Algebraic category of $LM_{n}$ 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 ($LM_{n}$) logic algebras were defined axiomatically in 1970, in ref. , as N-valued logic algebra representations and extensions of the \L ukasiewcz (3-valued) logics; then, the universal properties of categories of $LM_{n}$ -logic algebras were also investigated and reported in a series of recent publications ( and references cited therein). Recently, several modifications of $LM_{n}$-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 (), and in single-cell organisms, or in tumor growth. For a recent review on $n$-valued logic algebras, and major published results, the reader is referred to .

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

\rm

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

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

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

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

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

\end{exe}

Generalized logic spaces defined by $LM_{n}$ algebraic logics

• topological semigroup spaces of topological automata topological groupoid spaces of reset automata modules

Applications of generalized topoi:

• Modern quantum logic (MQL)
• Generalized quantum automata
• Mathematical models of N-state genetic networks
• Mathematical models of parallel computing networks