0.1 Introduction
Generalized topoi (toposes) with many-valued algebraic logic subobject classifiers are specified by
the associated categories 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 infinity,
etc.
0.2 Algebraic category of LMn 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
(LMn) 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 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 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].
0.3 Generalized logic spaces defined by LMn algebraic logics
0.4 Axioms defining generalized topoi
- Consider a subobject logic classifier Ω defined as an LM-algebraic logic Ln in the
category L of LM-logic algebras, together with logic-valued functors Fω : L → V ,
where V is the class of N logic values, with N needing not be finite.
- A triple (Ω,L,Fω) defines a generalized topos, τ, if the above axioms defining Ω are
satisfied, and if the functor Fω is an univalued functor in the sense of Mitchell.
More to come...
0.5 Applications of generalized topoi:
0.6 Applications of generalized topoi:
0.7 Generalized logic ‘spaces’ defined by LMn.
References
[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.
