PlanetPhysics/Generalized Toposes With Many Valued Logic Subobject Classifiers

Introduction
Generalized topoi (toposes) with many-valued \htmladdnormallink{algebraic {http://planetphysics.us/encyclopedia/CoIntersections.html} 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 LMn logic algebras
\L{}ukasiewicz 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. \L{}ukasiewicz-Moisil ($$LM_n$$) logic algebras were defined axiomatically in 1970, in ref. , as N-valued logic algebra representations and extensions of the \L ukasiewcz (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 ( 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, 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.

Generalized logic spaces defined by LMn algebraic logics

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

Axioms defining generalized topoi

 * Consider a subobject logic classifier $$\Omega$$ defined as an LM-algebraic logic $$L_n$$ in the category $${\mathbf L}$$ of LM-logic algebras, together with logic-valued functors $$F_{\omega}: {\mathbf L} \to V$$, where $$V$$ is the class of N logic values, with $$N$$ needing not be finite.
 * A triple $$(\Omega,L,F_{\omega})$$ defines a generalized topos, $$\tau$$, if the above axioms defining $$\Omega$$ are satisfied, and if the functor $$F_{\omega}$$ is an univalued functor in the sense of Mitchell.

{\mathbf More to come...}

Applications of generalized topoi:

 * Modern quantum logic (MQL)
 * Generalized quantum automata
 * Mathematical models of N-state genetic networks
 * Mathematical models of parallel computing networks

Applications of generalized topoi:

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Generalized logic `spaces' defined by LMn.

 * XY
 * YZ