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generalized toposes with many-valued logic subobject classifiers
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Generalized topoi (toposes) with many-valued algebraic 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.
More to come...
 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.  ukasiewicz-Moisil ($LM_n$ ) logic algebras were defined axiomatically in 1970, in ref. [1], as n-valued logic algebra representations and extensions of the  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 ([2] 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 ([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 $\mathcal{LM}$ of  ukasiewicz-Moisil, $n$ -valued logic algebras ($LM_n$ ), and $LM_n$ -lattice morphisms, $\lambda_{LM_n}$ , was introduced in 1970 in ref. [1] as an algebraic category tool for $n$ -valued logic studies. The objects of $\mathcal{LM}$ are the non-commutative $LM_n$ lattices and the morphisms of $\mathcal{LM}$ are the $LM_n$ -lattice morphisms as defined next.
Definition 1.1
A $n$ -valued  ukasiewicz-Moisil algebra, ($LM_{n}$ -algebra) is a structure of the form $(L,\vee,\wedge,N,(\phii)_{i\in\{1,\ldots,n-1\}},0,1)$ , 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 $i\in\{1,\ldots,n-1\}$ , $\phii:L\lra L$ is a lattice endomorphism;
- (L3) For each $i\in\{1,\ldots,n-1\},x\in L$ , $\phii(x)\vee N{\phii(x)}=1$ and $\phii(x)\wedge N{\phii(x)}=0$ ;
- (L4) For each $i,j\in\{1,\ldots,n-1\}$ , $\phii\circ\phi_{j}=\phi_{k}$ iff $(i+j)= k$ ;
- (L5) For each $i,j\in\{1,\ldots,n-1\}$ , $i\leq j$ implies $\phii\leq\phi_{j}$ ;
- (L6) For each $i\in\{1,\ldots,n-1\}$ and $x\in L$ , $\phii(N x)=N\phi_{n-i}(x)$ .
- (L7) Moisil's `determination principle': $$\left[\orc i\in\{1,\ldots,n-1\},\;\phii(x)=\phii(y)\right] \; implies \; [x = y] \;$$ [1,2].
Example 1.1 Let $L_n=\{0,1/(n-1),\ldots,(n-2)/(n-1),1\}$ . This set can be naturally endowed with an $\mbox{LM}_n$ -algebra structure as follows:
Note that, for $n=2$ , $L_n=\{0,1\}$ , and there is only one Chrysippian endomorphism of $L_n$ is $\phi_1$ , which is necessarily restricted by the determination principle to a bijection, thus making $L_n$ a Boolean algebra (if we were also to disregard the redundant bijection $\phi_1$ ). Hence, the `overloaded' notation $L_2$ , which is used for both the classical Boolean algebra and the two-element $\mbox{LM}_2$ -algebra, remains consistent.
Example 1.2 Consider a Boolean algebra $(B,\v,\w,{}^-,0,1)$ . Let $T(B)=\{(x_1,\ldots,x_n)\in B^{n-1}\mid x_1\leq\ldots\leq x_{n-1}\}$ . On the set $T(B)$ , we define an $\mbox{LM}_n$ -algebra structure as follows:
- the lattice operations, as well as $0$ and $1$ , are defined component-wise from $\Ld$ ;
- for each $(x_1,\ldots,x_{n-1})\in T(B)$ and $i\in\{1,\ldots,n-1\}$ one has:
$N(x_1,\ldots x_{n-1})=(\ov{x_{n-1}},\ldots,\ov{x_1})$ and $\phii(x_1,\ldots,x_n)=(x_i,\ldots,x_i) .$
- 1
- Georgescu, G. and C. Vraciu. 1970, On the characterization of centered
 ukasiewicz algebras., J. Algebra, 16: 486-495.
- 2
- Georgescu, G. 2006, N-valued Logics and
 ukasiewicz-Moisil Algebras, Axiomathes, 16 (1-2): 123-136.
- 3
- Baianu, I.C.: 1977, A Logical Model of Genetic Activities in
 ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology, 39: 249-258.
- 4
- Baianu, I.C.: 2004a.
 ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints-Sussex Univ.
- 5
- Baianu, I.C.: 2004b
 ukasiewicz-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
 ukasiewicz 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
 ukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1-2: 65-122.
Footnotes
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- The $\phii$ 's are called the Chrysippian endomorphisms of $L$ .
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See Also: non-Abelian structures, abelian category, supplemental axioms for an Abelian category, higher dimensional generalized Van Kampen theorems (HD-VKT), axiomatic theory of supercategories and metacategories, categorical quantum logics as quantum LM-algebraic logic, non-commuting graph, non-Abelian structures, quantum logic toposes
| Other names: |
LMn-algebraic n-valued logic, algebraic category of  logic algebras |
| Also defines: |
many-valued logic subobject classifier, algebraic category of LMn logic algebras |
| Keywords: |
generalized topoi with many-valued logic subobject classifiers, the category of n-valued, LMn-logic algebras and LMn-lattice morphisms, n-valued logic algebra, algebraic catgeory of n-valued logic lattices and lattice-morphisms |
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Cross-references: parallel, genetic networks, quantum logic, modules, topological groupoid, automata, semigroup, lattice, consistent, overloaded, redundant, Boolean algebra, bijection, rational numbers, order, induced, operations, bounded lattice, implies, iff, lattice endomorphism, property, involution, decreasing, distributive lattice, bounded, De Morgan algebra, axioms, structure, morphisms, objects, logic algebras, growth, genetic nets, biodynamics, quantum logics, valid, modifications, references, series, universal properties, extensions, representations, algebra, circuits, many-valued logics, infinity, cardinal, non-commutative, categories, subobject classifiers, logic, algebraic, topoi
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This is version 36 of generalized toposes with many-valued logic subobject classifiers, born on 2008-07-16, modified 2009-06-22.
Object id is 10803, canonical name is GeneralizedToposesTopoiWithManyValuedLogicSubobjectClassifiers.
Accessed 1252 times total.
Classification:
| AMS MSC: | 03B50 (Mathematical logic and foundations :: General logic :: Many-valued logic) | | | 03G20 (Mathematical logic and foundations :: Algebraic logic :: Lukasiewicz and Post algebras) | | | 03G30 (Mathematical logic and foundations :: Algebraic logic :: Categorical logic, topoi) | | | 03B15 (Mathematical logic and foundations :: General logic :: Higher-order logic and type theory) | | | 18B25 (Category theory; homological algebra :: Special categories :: Topoi) | | | 58A03 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Topos-theoretic approach to differentiable manifolds) |
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Pending Errata and Addenda
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