generalized toposes with many-valued logic subobject classifiers

generalized toposes with many-valued logic subobject classifiers

\xyoption

curve

1

Generalized toposes

1.1

Introduction

Generalized topoi (toposes) with many-valued algebraic logic

subobject classifiers

are 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

L

⁢

M

of Łukasiewicz-Moisil,

n

-valued logic algebras (

L

⁢

M

n

), and

L

⁢

M

n

–

lattice

morphisms

,

λ

L

⁢

M

n

, was introduced in 1970 in ref.

[

1

]

as an

algebraic category

tool for

n

-valued logic studies. The objects of

ℒ

⁢

ℳ

are the

non–

commutative

L

⁢

M

n

lattices and the morphisms of

ℒ

⁢

ℳ

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

,

∨

,

∧

,

N

,

(

φ

i

)

i

∈

{

1

,

…

,

n

-

1

}

,

0

,

1

)

, subject to the following axioms:

•

(L1)

(

L

,

∨

,

∧

,

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

∨

y

)

=

N

⁢

x

∧

N

⁢

y

;

•

(L2) For each

i

∈

{

1

,

…

,

n

-

1

}

,

φ

i

:

L

⟶

L

is a

lattice endomorphism

;

$*$

$*$

The

φ

i

’s are called the

Chrysippian

endomorphisms

of

L

.

•

(L3) For each

i

∈

{

1

,

…

,

n

-

1

}

,

x

∈

L

,

φ

i

⁢

(

x

)

∨

N

⁢

φ

i

⁢

(

x

)

=

1

and

φ

i

⁢

(

x

)

∧

N

⁢

φ

i

⁢

(

x

)

=

0

;

•

(L4) For each

i

,

j

∈

{

1

,

…

,

n

-

1

}

,

φ

i

∘

φ

j

=

φ

k

iff

(

i

+

j

)

=

k

;

•

(L5) For each

i

,

j

∈

{

1

,

…

,

n

-

1

}

,

i

⩽

j

implies

φ

i

⩽

φ

j

;

•

(L6) For each

i

∈

{

1

,

…

,

n

-

1

}

and

x

∈

L

,

φ

i

⁢

(

N

⁢

x

)

=

N

⁢

φ

n

-

i

⁢

(

x

)

.

•

(L7) Moisil’s ‘determination principle’:

[

∀

i

∈

{

1

,

…

,

n

-

1

}

,

φ

i

(

x

)

=

φ

i

(

y

)

]

i

m

p

l

i

e

s

[

x

=

y

]

[

1

,

2

]

.

Example 1.1

.

Let

L

n

=

{

0

,

1

/

(

n

-

1

)

,

…

,

(

n

-

2

)

/

(

n

-

1

)

,

1

}

. This set can be naturally endowed with an

LM

n

–algebra structure as follows:

•

the

bounded lattice

operations

are those

induced

by the usual order on

rational numbers

;

•

for each

j

∈

{

0

,

…

,

n

-

1

}

,

N

⁢

(

j

/

(

n

-

1

)

)

=

(

n

-

j

)

/

(

n

-

1

)

;

•

for each

i

∈

{

1

,

…

,

n

-

1

}

and

j

∈

{

0

,

…

,

n

-

1

}

,

φ

i

⁢

(

j

/

(

n

-

1

)

)

=

0

if

j

<

i

and

=

1

otherwise.

Note that, for

n

=

2

,

L

n

=

{

0

,

1

}

, and there is only one Chrysippian endomorphism of

L

n

is

φ

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

φ

1

). Hence, the ‘overloaded’ notation

L

2

, which is
used for both the classical Boolean algebra and the two–

element

LM

2

–algebra, remains

consistent

.

Example 1.2

.

Consider a Boolean algebra

B

.

Let

T

⁢

(

B

)

=

{

(

x

1

,

…

,

x

n

)

∈

B

n

-

1

∣

x

1

⩽

…

⩽

x

n

-

1

}

. On the set

T

⁢

(

B

)

, we define an

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

,

…

,

x

n

-

1

)

∈

T

⁢

(

B

)

and

i

∈

{

1

,

…

,

n

-

1

}

one has:

N

⁢

(

x

1

,

…

⁢

x

n

-

1

)

=

(

x

n

-

1

¯

,

…

,

x

1

¯

)

and

φ

i

⁢

(

x

1

,

…

,

x

n

)

=

(

x

i

,

…

,

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

→

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,

τ

, if the above axioms
defining

O

are satisfied, and if the functor

F

⁢

o

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

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

,

http://en.wikipedia.org/wiki/User:Bci2/Books/Interactomics

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.

8

Mitchell, Barry. The Theory of Categories. Academic Press: London, 1968.

Title

generalized toposes with many-valued logic subobject classifiers

Canonical name

GeneralizedToposesWithManyvaluedLogicSubobjectClassifiers

Date of creation

2013-03-22 18:13:11

Last modified on

2013-03-22 18:13:11

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

60

Author

bci1 (20947)

Entry type

Topic

Classification

msc 58A03

Classification

msc 18B25

Classification

msc 03B15

Classification

msc 03G30

Classification

msc 03G20

Classification

msc 03B50

Synonym

LMn-algebraic n-valued logic

Synonym

algebraic category of

L

⁢

M

n

logic algebras

Related topic

NonAbelianStructures

Related topic

AbelianCategory

Related topic

AxiomsForAnAbelianCategory

Related topic

GeneralizedVanKampenTheoremsHigherDimensional

Related topic

AxiomaticTheoryOfSupercategories

Related topic

CategoricalOntology

Related topic

NonCommutingGraphOfAGroup

Related topic

NonAbelianStructures

Related topic

QuantumLogicsTopoi

Related topic

CategoricalAlgebr

Defines

many-valued logic subobject classifier

Defines

algebraic category of LMn logic algebras

Defines

noncommutative lattice

Generated on Sat Feb 10 11:49:48 2018 by

LaTeXML