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 [[../Cod/|categories]] of algebraic logics previously defined as $L{M}_n$, that is, [[../AbelianCategory3/|non-commutative]] lattices with $n$ logical values, where $n$ can also be chosen to be any cardinal, including infinity, etc.

Algebraic category of LM_n logic algebras

\L{}ukasiewicz logic algebras were constructed by Grigore Moisil in 1941 to define `nuances' in logics, or [[../LM_nLogicAlgebra/|many-valued logics]], as well as 3-state control logic (electronic) circuits. \L{}ukasiewicz-Moisil ($L{M}_n$) logic algebras were defined axiomatically in 1970, in ref. , as [[../LM_nLogicAlgebra/|N-valued logic algebra]] [[../CategoricalGroupRepresentation/|representations]] and extensions of the \L ukasiewcz (3-valued) logics; then, the universal properties of categories of $L{M}_n$ -logic algebras were also investigated and reported in a series of recent publications ( and references cited therein). Recently, several modifications of $L{M}_n$-logic algebras are under consideration as valid candidates for representations of [[../LQG2/|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 LM_n algebraic logics

• [[../CoIntersections/|topological]] [[../TrivialGroupoid/|semigroup]] spaces of topological automata [[../GroupoidHomomorphism2/|topological groupoid]] spaces of reset automata [[../RModule/|modules]]

Axioms defining generalized topoi

• Consider a subobject logic classifier $\mathrm{\Omega }$ defined as an LM-algebraic logic $L_n$ in the category $𝐋$ of LM-logic algebras, together with logic-valued [[../TrivialGroupoid/|functors]] $F_{\omega }:𝐋\to V$, where $V$ is the class of N logic values, with $N$ needing not be finite.
• A triple $(\mathrm{\Omega },L,F_{\omega })$ defines a generalized [[../GrothendieckTopos/|topos]], $\tau$, if the above axioms defining $\mathrm{\Omega }$ 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 [[../QuantumComputers/|quantum automata]]
• Mathematical models of N-state [[../GeneNetDigraph/|genetic networks]]
• Mathematical models of parallel computing networks

Applications of generalized topoi:

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Generalized logic `spaces' defined by LMn.

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