PlanetPhysics/Generalized Topoi With LMn Algebraic Logic Classifiers

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 $$\mathcal{LM'' $$ 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 $$(L,\vee,\wedge,N,(\phii)_{i\in\{1,\ldots,n-1\}},0,1)$$, 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 $$i\in\{1,\ldots,n-1\}$$, $$\phii:L\lra L$$ is a lattice endomorphism;\footnote{ The $$\phii$$'s are called the Chrysippian endomorphisms of $$L$$.}
 * (L3) For each $$i\in\{1,\ldots,n-1\},x\in L$$, $$\phii(x)\vee N{\phii(x)}=1$$ and $$\phii(x)\wedge N{\phii(x)}=0$$;
 * (L4) For each $$i,j\in\{1,\ldots,n-1\}$$, $$\phii\circ\phi_{j}=\phi_{k}$$ iff $$(i+j)= k$$;
 * (L5) For each $$i,j\in\{1,\ldots,n-1\}$$, $$i\leq j$$ implies $$\phii\leq\phi_{j}$$;
 * (L6) For each $$i\in\{1,\ldots,n-1\}$$ and $$x\in L$$, $$\phii(N x)=N\phi_{n-i}(x)$$.
 * (L7) Moisil's `determination principle': $$\left[\orc i\in\{1,\ldots,n-1\},\;\phii(x)=\phii(y)\right] \; implies \; [x = y] \;$$.

\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:

\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 $$(B,\v,\w,{}^-,0,1)$$. Let $$T(B)=\{(x_1,\ldots,x_n)\in B^{n-1}\mid x_1\leq\ldots\leq x_{n-1}\}T(B)\mbox{LM}_n$$-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 $$j\in\{0,\ldots,n-1\}$$, $$\phii(j/(n-1))=0$$ if $$j<i$$ and $$=1$$ otherwise.

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

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