generalized toposes with many-valued logic subobject classifiers

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1 Generalized toposes

1.1 Introduction

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

1.2 Algebraic category of $L{M}_n$ 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 ($L{M}_n$) 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 $L{M}_n$ -logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of $L{M}_n$-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 ([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].

The category $\mathrm{L}\mathit{}\mathrm{M}$ of Łukasiewicz-Moisil, $n$-valued logic algebras ($L\mathit{}{M}_n$), and $L\mathit{}{M}_n$–lattice morphisms, ${\lambda }_{L{M}_n}$, was introduced in 1970 in ref. [1] as an algebraic category tool for $n$-valued logic studies. The objects of $ℒℳ$ are the non–commutative $L{M}_n$ lattices and the morphisms of $ℒℳ$ are the $L{M}_n$-lattice morphisms as defined next.

Definition 1.1.

A $n$–valued Łukasiewicz–Moisil algebra, ($L{M}_n$–algebra) is a structure of the form$(L,\vee ,\wedge ,N,{({\phi }_i)}_{i\in \{1,\mathrm{\dots },n-1\}},0,1)$, subject to the following axioms:

• (L1) $(L,\vee ,\wedge ,N,0,1)$ is a de Morgan algebra, that is, a bounded distributive lattice with a decreasing involution $N$ satisfying the de Morgan property $N(x\vee y)=Nx\wedge Ny$;

• (L2) For each $i\in \{1,\mathrm{\dots },n-1\}$, ${\phi }_i:L⟶L$ is a lattice endomorphism;

• (L3) For each $i\in \{1,\mathrm{\dots },n-1\},x\in L$, ${\phi }_i(x)\vee N{\phi }_i(x)=1$ and${\phi }_i(x)\wedge N{\phi }_i(x)=0$;

• (L4) For each $i,j\in \{1,\mathrm{\dots },n-1\}$, ${\phi }_i\circ {\phi }_j={\phi }_k$ iff $(i+j)=k$;

• (L5) For each $i,j\in \{1,\mathrm{\dots },n-1\}$, $i⩽j$ implies ${\phi }_i⩽{\phi }_j$;

• (L6) For each $i\in \{1,\mathrm{\dots },n-1\}$ and $x\in L$, ${\phi }_i(Nx)=N{\phi }_{n-i}(x)$.

• (L7) Moisil’s ‘determination principle’:

  $$[\forall i\in \{1,\mathrm{\dots },n-1\},{\phi }_i(x)={\phi }_i(y)]implies[x=y]$$

  [1, 2].

Example 1.1.

Let $L_n=\{0,1/(n-1),\mathrm{\dots },(n-2)/(n-1),1\}$. This set can be naturally endowed with an ${\text{LM}}_n$–algebra structure as follows:

• the bounded lattice operations are those induced by the usual order on rational numbers;

• for each $j\in \{0,\mathrm{\dots },n-1\}$, $N(j/(n-1))=(n-j)/(n-1)$;

• for each $i\in \{1,\mathrm{\dots },n-1\}$ and $j\in \{0,\mathrm{\dots },n-1\}$,${\phi }_i(j/(n-1))=0$ if and $=1$ otherwise.

Note that, for $n=2$, $L_n=\{0,1\}$, and there is only one Chrysippian endomorphism of $L_n$ is ${\phi }_1$, whichis necessarily restricted by the determination principle to a bijection, thus making $L_n$ a Boolean algebra (ifwe were also to disregard the redundant bijection ${\phi }_1$). Hence, the ‘overloaded’ notation $L_2$, which isused for both the classical Boolean algebra and the two–element ${\text{LM}}_2$–algebra, remains consistent.

Example 1.2.

Consider a Boolean algebra $B$.

Let $T(B)=\{(x_1,\mathrm{\dots },x_n)\in B^{n-1}\mid x_1⩽\mathrm{\dots }⩽x_{n-1}\}$. On the set $T(B)$, we define an ${\text{LM}}_n$-algebra structure as follows:

• the lattice operations, as well as $0$ and $1$, are defined component–wise from $L_2$;

• for each $(x_1,\mathrm{\dots },x_{n-1})\in T(B)$ and $i\in \{1,\mathrm{\dots },n-1\}$ one has:
  $N(x_1,\mathrm{\dots }{x}_{n-1})=(\overline{x_{n-1}},\mathrm{\dots },\overline{x_1})$ and ${\phi }_i(x_1,\mathrm{\dots },x_n)=(x_i,\mathrm{\dots },x_i).$

1.3 Generalized logic spaces defined by $L{M}_n$ algebraic logics

• Topological semigroup spaces of topological automata

• Topological groupoid spaces of reset automata modules

1.4 Axioms defining a generalized topos

• Consider a subobject logic classifier $O$ defined as an LM-algebraic logic$L_n$ in the category L of LM-logic algebras, together with logic-valued functors $F_o:L\to V$,where $V$ is the class of N logic values, with $N$ needing not be finite.

• A triple $(O,𝐋,F_o)$ defines a generalized topos, $\tau$, if the above axiomsdefining $O$ are satisfied, and if the functor $Fo$ is an univalued functor in the sense of Mitchell.

More to come…

1.5 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