generalized toposes with many-valued logic subobject classifiers\xyoption
1 Generalized toposes
1.2 Algebraic category of 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 () logic algebras were defined axiomatically in 1970, in ref. , as n-valued logic algebra representations and extensions of the Łukasiewcz (3-valued) logics; then, the universal properties of categories of -logic algebras were also investigated and reported in a series of recent publications ( and references cited therein). Recently, several modifications of -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 -valued logic algebras, and major published results, the reader is referred to .
The category of Łukasiewicz-Moisil, -valued logic algebras (), and –lattice morphisms, , was introduced in 1970 in ref.  as an algebraic category tool for -valued logic studies. The objects of are the non–commutative lattices and the morphisms of are the -lattice morphisms as defined next.
(L3) For each , and ;
(L4) For each , iff ;
(L5) For each , implies ;
(L6) For each and , .
Note that, for , , and there is only one Chrysippian endomorphism of is , which is necessarily restricted by the determination principle to a bijection, thus making a Boolean algebra (if we were also to disregard the redundant bijection ). Hence, the ‘overloaded’ notation , which is used for both the classical Boolean algebra and the two–element –algebra, remains consistent.
Consider a Boolean algebra .
Let . On the set , we define an -algebra structure as follows:
the lattice operations, as well as and , are defined component–wise from ;
for each and one has:
1.3 Generalized logic spaces defined by algebraic logics
1.4 Axioms defining a generalized topos
A triple defines a generalized topos, , if the above axioms defining are satisfied, and if the functor is an univalued functor in the sense of Mitchell.
More to come…
1.5 Applications of generalized topoi:
- 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, http://en.wikipedia.org/wiki/User:Bci2/Books/InteractomicsAbstract 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.
- 8 Mitchell, Barry. The Theory of Categories. Academic Press: London, 1968.
|Title||generalized toposes with many-valued logic subobject classifiers|
|Date of creation||2013-03-22 18:13:11|
|Last modified on||2013-03-22 18:13:11|
|Last modified by||bci1 (20947)|
|Synonym||LMn-algebraic n-valued logic|
|Synonym||algebraic category of logic algebras|
|Defines||many-valued logic subobject classifier|
|Defines||algebraic category of LMn logic algebras|