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generalized topoi with LMn algebraic logic classifiers (Topic)

Generalized toposes

Introduction

Generalized topoi (toposes) with many-valued algebraic 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

Łukasiewicz 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. Łukasiewicz-Moisil ($LM_n$) logic algebras were defined axiomatically in 1970, in ref. [1], as N-valued logic algebra representations and extensions of the Łukasiewcz (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 ([2] 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 ([3]), 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 [2].

The category $\mathcal{LM}$ of Łukasiewicz-Moisil, $n$-valued logic algebras ($LM_n$), and $LM_n$–lattice morphisms, $\lambda_{LM_n}$, was introduced in 1970 in ref. [1] 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.

Definition 1.1  

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

  • (L1) $(L,\vee,\wedge,N,0,1)$ is a 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 $i\in\{1,\ldots,n-1\}$, $\phi _{i}:L{\longrightarrow}L$ is a lattice endomorphism; % latex2html id marker 2293 $ \fnsymbol{footnote}$
  • (L3) For each $i\in\{1,\ldots,n-1\},x\in L$, $\phi _{i}(x)\vee N{\phi _{i}(x)}=1$ and $\phi _{i}(x)\wedge N{\phi _{i}(x)}=0$;
  • (L4) For each $i,j\in\{1,\ldots,n-1\}$, $\phi _{i}\circ\phi _{j}=\phi _{k}$ iff $(i+j)= k$;
  • (L5) For each $i,j\in\{1,\ldots,n-1\}$, $i\leqslant j$ implies $\phi _{i}\leqslant \phi _{j}$;
  • (L6) For each $i\in\{1,\ldots,n-1\}$ and $x\in L$, $\phi _{i}(N x)=N\phi _{n-i}(x)$.
  • (L7) Moisil's `determination principle':

    $\displaystyle \left[\forall i\in\{1,\ldots,n-1\},\;\phi _{i}(x)=\phi _{i}(y)\right] \; implies \; [x = y] \;$
    [1,2].
Example 1.1   Let $L_n=\{0,1/(n-1),\ldots,(n-2)/(n-1),1\}$. This set can be naturally endowed with an 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 $j\in\{0,\ldots,n-1\}$, $\phi _{i}(j/(n-1))=0$ if $j<i$ and $=1$ otherwise.
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 LM$_2$–algebra, remains consistent.
Example 1.2   Consider a Boolean algebra $(B,\v ,\wedge ,{}^-,0,1)$. Let $T(B)=\{(x_1,\ldots,x_n)\in B^{n-1}\mid x_1\leqslant \ldots\leqslant x_{n-1}\}$. On the set $T(B)$, we define an LM$_n$-algebra structure as follows:
  • the lattice operations, as well as 0 and $1$, are defined component–wise from $L_{2}$;
  • for each $(x_1,\ldots,x_{n-1})\in T(B)$ and $i\in\{1,\ldots,n-1\}$ one has:
    $N(x_1,\ldots x_{n-1})=(\overline {x_{n-1}},\ldots,\overline {x_1})$ and $\phi _{i}(x_1,\ldots,x_n)=(x_i,\ldots,x_i) .$

Generalized logic spaces defined by $LM_n$ algebraic logics

Applications of generalized topoi:

Bibliography

1
Georgescu, G. and C. Vraciu. 1970, On the characterization of centered Łukasiewicz algebras., J. Algebra, 16: 486-495.
2
Georgescu, G. 2006, N-valued Logics and Łukasiewicz-Moisil Algebras, Axiomathes, 16 (1-2): 123-136.
3
Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology, 39: 249-258.
4
Baianu, I.C.: 2004a. Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints–Sussex Univ.
5
Baianu, I.C.: 2004b Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).
6
Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional Algebra, Abstract and Preprint of Report in PDF .
7
Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1–2: 65–122.


Footnotes

... endomorphism; % latex2html id marker 2293 $ \fnsymbol{footnote}$
The $\phi_{i}$'s are called the Chrysippian endomorphisms of $L$.


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Cross-references: genetic networks, quantum automata, modules, topological groupoid, semigroup, topological, operations, morphisms, objects, algebraic category, quantum logics, representations, N-valued logic algebra, many-valued logics, non-commutative, categories, algebraic

This is version 2 of generalized topoi with LMn algebraic logic classifiers, born on 2009-06-21, modified 2009-06-22.
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 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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