generalized toposes with many-valued logic subobject classifiers



1 Generalized toposes

1.1 Introduction

Generalized topoi (toposes) with many-valued algebraic logic subobject classifiers are specified by the associated categoriesMathworldPlanetmath of algebraic logics previously defined as LMn, that is, non-commutative lattices with n logical values, where n can also be chosen to be any cardinal, including infinityMathworldPlanetmath, etc.

1.2 Algebraic category of LMn logic algebras

Łukasiewicz logic algebrasPlanetmathPlanetmath were constructed by Grigore Moisil in 1941 to define ‘nuances’ in logics, or many-valued logics, as well as 3-state control logic (electronic) circuitsMathworldPlanetmath. Łukasiewicz-Moisil (LMn) logic algebras were defined axiomatically in 1970, in ref. [1], as n-valued logic algebra representations and extensionsPlanetmathPlanetmathPlanetmath of the Łukasiewcz (3-valued) logics; then, the universal propertiesMathworldPlanetmath of categories of LMn -logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of LMn-logic algebras are under consideration as valid candidates for representations of quantum logicsPlanetmathPlanetmath, 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 LM of Łukasiewicz-Moisil, n-valued logic algebras (LMn), and LMnlatticeMathworldPlanetmath morphismsMathworldPlanetmath, λLMn, was introduced in 1970 in ref. [1] as an algebraic categoryPlanetmathPlanetmathPlanetmath tool for n-valued logic studies. The objects of are the non–commutativePlanetmathPlanetmath LMn lattices and the morphisms of are the LMn-lattice morphisms as defined next.

Definition 1.1.

A n–valued Łukasiewicz–Moisil algebraMathworldPlanetmath, (LMn–algebra) is a structureMathworldPlanetmath of the form (L,,,N,(φi)i{1,,n-1},0,1), subject to the following axioms:

  • (L1) (L,,,N,0,1) is a de Morgan algebra, that is, a bounded distributive latticeMathworldPlanetmath with a decreasing involutionPlanetmathPlanetmath N satisfying the de Morgan property N(xy)=NxNy;

  • (L2) For each i{1,,n-1}, φi:LL is a lattice endomorphismMathworldPlanetmath;$*$$*$The φi’s are called the Chrysippian endomorphismsMathworldPlanetmathPlanetmathPlanetmath of L.

  • (L3) For each i{1,,n-1},xL, φi(x)Nφi(x)=1 and φi(x)Nφi(x)=0;

  • (L4) For each i,j{1,,n-1}, φiφj=φk iff (i+j)=k;

  • (L5) For each i,j{1,,n-1}, ij implies φiφj;

  • (L6) For each i{1,,n-1} and xL, φi(Nx)=Nφn-i(x).

  • (L7) Moisil’s ‘determination principle’:


    [1, 2].

Example 1.1.

Let Ln={0,1/(n-1),,(n-2)/(n-1),1}. This set can be naturally endowed with an LMn –algebra structure as follows:

  • the bounded latticeMathworldPlanetmath operationsMathworldPlanetmath are those induced by the usual order on rational numbers;

  • for each j{0,,n-1}, N(j/(n-1))=(n-j)/(n-1);

  • for each i{1,,n-1} and j{0,,n-1}, φi(j/(n-1))=0 if j<i and =1 otherwise.

Note that, for n=2, Ln={0,1}, and there is only one Chrysippian endomorphism of Ln is φ1, which is necessarily restricted by the determination principle to a bijection, thus making Ln a Boolean algebraMathworldPlanetmath (if we were also to disregard the redundant bijection φ1). Hence, the ‘overloaded’ notation L2, which is used for both the classical Boolean algebra and the two–element LM2–algebra, remains consistentPlanetmathPlanetmath.

Example 1.2.

Consider a Boolean algebra B.

Let T(B)={(x1,,xn)Bn-1x1xn-1}. On the set T(B), we define an LMn-algebra structure as follows:

  • the lattice operations, as well as 0 and 1, are defined componentMathworldPlanetmathPlanetmathPlanetmath–wise from L2;

  • for each (x1,,xn-1)T(B) and i{1,,n-1} one has:
    N(x1,xn-1)=(xn-1¯,,x1¯) and φi(x1,,xn)=(xi,,xi).

1.3 Generalized logic spaces defined by LMn algebraic logics

1.4 Axioms defining a generalized topos

  • Consider a subobject logic classifier O defined as an LM-algebraic logic Ln in the category L of LM-logic algebras, together with logic-valued functorsMathworldPlanetmath Fo:LV, where V is the class of N logic values, with N needing not be finite.

  • A triple (O,𝐋,Fo) defines a generalized topos, τ, if the above axioms defining O are satisfied, and if the functor Fo is an univalued functor in the sense of Mitchell.

More to come…

1.5 Applications of generalized topoi:

  • Modern quantum logic (MQL)

  • Generalized quantum automata

  • Mathematical models of N-state genetic networks [7]

  • Mathematical models of parallel computing networks


  • 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 RelationMathworldPlanetmathPlanetmathPlanetmath to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional AlgebraPlanetmathPlanetmath, 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: TransformationsPlanetmathPlanetmath of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1–2: 65–122.
  • 8 Mitchell, Barry. The Theory of Categories. Academic Press: London, 1968.
Title generalized toposes with many-valued logic subobject classifiers
Canonical name GeneralizedToposesWithManyvaluedLogicSubobjectClassifiers
Date of creation 2013-03-22 18:13:11
Last modified on 2013-03-22 18:13:11
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 60
Author bci1 (20947)
Entry type Topic
Classification msc 58A03
Classification msc 18B25
Classification msc 03B15
Classification msc 03G30
Classification msc 03G20
Classification msc 03B50
Synonym LMn-algebraic n-valued logic
Synonym algebraic category of LMn logic algebras
Related topic NonAbelianStructures
Related topic AbelianCategory
Related topic AxiomsForAnAbelianCategory
Related topic GeneralizedVanKampenTheoremsHigherDimensional
Related topic AxiomaticTheoryOfSupercategories
Related topic CategoricalOntology
Related topic NonCommutingGraphOfAGroup
Related topic NonAbelianStructures
Related topic QuantumLogicsTopoi
Related topic CategoricalAlgebr
Defines many-valued logic subobject classifier
Defines algebraic category of LMn logic algebras
Defines noncommutative lattice