Introduction

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

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. [1], 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 ([2] 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 ([3]), 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 [2].

Generalized logic spaces defined by algebraic logics

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

Axioms defining generalized topoi

• Consider a subobject logic classifier defined as an LM-algebraic logic in the category of LM-logic algebras, together with logic-valued functors , where is the class of N logic values, with needing not be finite.
• 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...

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

Applications of generalized topoi:

• XY
• YZ

Generalized logic `spaces' defined by LMn.

• XY
• YZ

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.