al from an alysidal set Aal to an alysidal set Bal with the property that, for every l ..... b) Distributive property with respect to a sum of scalars: (λâ µ)⢠k n. â = λ⢠k n.
AN INTRODUCTION TO ALYSIDAL ALGEBRA (II) J. Nescolarde-Selva and F. Vives-Maciá Departamento de Matemática Aplicada. Universidad de Alicante. Alicante. Spain. J.L. Usó-Doménech Departament de Matemátiques. Universitat Jaume I. Castelló de la Plana. Spain. D. Berend Depts. of Math and of CS’. Ben-Gurion University. Beer-Sheva. Israel. Abstract
or :
jo
su e
.s el
va @
ua .
es
Purpose Deontical Impure Systems are systems whose object set is formed by an s-impure set, whose elements are perceptuales significances (relative beings) of material and/or energetic objects (absolute beings) and whose relational set is freeways of relations, formed by sheaves of relations going in two-way directions. Objects and freeways form chains. Design/methodology/approach – Mathematical and logical development of human society structure. Findings – Existence of relations with positive imperative modality (obligation) would constitute the skeleton of the system. Negative imperative modality (prohibition) would be the immunological system of protection of the system. Modality permission the muscular system, that gives the necessary flexibility to the system, in as much to the modality faculty its neurocerebral system, because it allows him to make decisions. Transactions of energy, money, merchandise, population, etc., would be the equivalent one to the sanguineous system. These economic transactions and inferential relations, depend, as well, of the existence of a legislative body with their obligations, prohibitions and permissions that regulate them. Originality/ Value This paper is a continuation of the previous one, continuing the development of a Alysidal Algebra, which is important for the study of the Deontical Impure Systems. They are defined coupling functions and alysidal structures. It is defined a special coupling function denominated gnorpsic function that can be used for algebraic operations between alysidal sets. Keywords Alysidal set, Chains, Freeway of relations, Coupling functions, Sheaf of relations, Structural functions.
au th
1. COUPLING FUNCTIONS
po C stor p re rin sp t on di ng
1.1. Freeway theory
Definition 1: A coupling function
nxm
f : Aal → Bal is surjective iff its alysidal range al
nxm
f ( A ) is equal to its alysidal codomain Bal. al
al
A surjective coupling function is called a coupling surjection. 1) For any alysidal set Aal, the identity coupling function idA on A is surjective. There always exists a coupling function "reversible" by a coupling surjection. Every coupling function with a right inverse is a coupling surjection. 2) The converse is equivalent to the axiom of choice: a coupling function nxm
f : Aal → Bal
is a coupling surjective function iff there exists a function
al
nxm
nxm
g :Bal → Aal al
such that,
f al
nxm
o
g
equals the identity function on Bal. Note that
al
nxm
g
nxm
may not be a complete inverse of
al
f al
1
because the composition in the other
nxm
nxm
nxm
g f
order,
o
al
, may not be the identity on Aal. In other words,
f
can undo or
al
al nxm
g , but not necessarily can be reversed by it.
"reverse"
al
3) Coupling surjections are not always invertible (bijective coupling function). nxm
f
4) If
nxm
al nxm
al nxm
f
5) If
nxm
o
nxm
g are both surjective, then f
and
g
al
o
g
al
is a surjective coupling function.
al nxm
is a surjective coupling function, then
al
f
is a surjective coupling
al nxm
g
need not be).
es
function (but
al nxm
f : Aal → Bal
is a surjective couplng function iff, given any coupling functions
ua .
6)
al
nxm
nxm
g , h : Bal → C al , whenever g o
f
al
al
al
al
nxm
nxm
h
=
nxm
va @
nxm nxm
f
o
al
, then
al
g al
nxm
=
h.
In other
al
.s el
words, surjective coupling functions are precisely the epimorphism in the category Alysidal Set of alysidal sets. nxm
su e
f : Aal → Bal is a surjective coupling function and Bal’ is an alysidal subset
7) If
f al
nxm −1
f (B ' ) ) . al
(Bal ') = Bal ' . Thus, Bal’ can be recovered from its preimage
po C stor p re rin sp t on di ng
al
−1
or :
of Bal, then
nxm f al
au th
nxm
jo
al
8) For any coupling function
nxm
h: Aal →Cal
there exists a coupling surjection
al
nxm
f : Aal → Bal
nxm
al nxm
go f al
nxm
nxm
and a coupling injection
g :B al →Cal
such that
h= al
al
.
al
9) Every coupling surjection induces a coupling bijection defined on a quotient of its nxm
alysidal domain. More precisely, every coupling surjection
f : Aal → Bal can al
be factored as a projection followed by a coupling bijection as follows: Let Aal/ ≈ be the equivalence classes of the alysidal set Aal under the following
f (℘ ) = f (℘ ) . Equivalently, Aal/ ≈
nxm
equivalence relation: ℘kn ≈ ℘lm iff
nxm
k n
al
is
al nxm
the alysdal set of all preimages under
l m
f
. Let P( ≈ ) : Aal → Aal/ ≈ be the
al
projection map which sends each ℘kn in Aal to its equivalence class [℘kn ]~, and let
2
nxm
f
: Aal /
al P nxm
by
→ Bal be the well-defined coupling function given
≈
f ([℘ ] ) = f (℘ ) . Then f nxm
k n
al
nxm
nxm
k n ≈
al
o P( ≈ ).
f
=
al P
al
nxm
10) If
f : Aal → Bal is a surjective coupling function, then Aal has at least as many al
alysidal elements as Bal, in the sense of cardinal numbers. 11) If both Aal and Bal are finite with the same number of alysidal elements, then nxm
f : Aal → Bal is a surjective coupling function iff f is an injective coupling al
ua .
nxm
Definition 2: A bijective coupling function is a coupling function
es
function.
f
from an alysidal
va @
al
set Aal to an alysidal set Bal with the property that, for every ℘lm in Bal, there is exactly
f (℘ ) = ℘
nxm
k n
l m
.
.s el
one ℘kn in Aal such that
su e
al
nxm
f
is a bijection coupling function if it is a one-to-one coupling
al
or :
correspondence between those alysidal sets.
jo
Alternatively,
au th
1) A bijective coupling function from an alysidal set to itself is also called an alysidal permutation.
po C stor p re rin sp t on di ng
2) The set of all bijection coupling functions from Aal to Bal is denoted as (AB)al. nxm −1
nxm
3) A coupling function
is bijective iff its inverse coupling relation
f
al
coupling function. In that case, nxm
4) The composition
g al
f
is a
nxm −1
f
is also a coupling bijection.
al
nxm
o
f al
nxm
of two coupling bijections
al
f
nxm
(AB)al and
g
al
is a coupling bijection. The inverse of
nxm
nxm
g
f
o
al
nxm
is ( g o
al
al
(BC)al
al nxm −1
nxm
f al
-1
) =
f al
nxm −1
o
g
.A
al
coupling bijection is composed of a coupling injection and a coupling surjection. nxm
On the other hand, if the composition
g al
3
nxm
o
f al
of two coupling functions is a
nxm
bijective coupling function, we can only say that
f
is an injective coupling
al
nxm
function and
g
is a surjective coupling function.
al nxm
5) A coupling relation
f
from Aal to Bal is a bijective coupling function iff there
al
nxm
nxm
g
from Bal to Aal such that
al
g
nxm
o
al nxm
coupling identity function on A, and
f
f
is the
al
nxm
o
al
g
is the coupling identity function
al
ua .
on B. Consequently, alysidal sets have the same cardinality.
es
exists another coupling relation
.s el
va @
6) If A and B are finite alysidal sets, then there exists a bijection coupling function between the two alysidal sets Aal and Bal iff Aal and Bal have the same number of alysidal elements.
au th
or :
jo
su e
7) For any alysidal set Aal, the identity coupling function idA from Aal to Bal, defined by idX(℘kn ) = ℘kn , is bijective coupling function. 8) For an alysidal subset Aal’ of the domain with cardinality |Aal’| and alysidal subset Bal’ of the codomain with cardinality |Bal’|, one has the following equalities: nxm
f ( A ') = al
Aal '
po C stor p re rin sp t on di ng
al
.
nxm −1
f ( B ') = B al
al
'
al
9) If Aal and Bal are finite alysidal sets with the same cardinality, and nxm
f : Aal → Bal , then the following are equivalent: al
nxm
1.
f
is a coupling bijection.
al
2.
nxm
f
is a coupling surjection.
al
nxm
3.
f
is a coupling injection.
al
10) At least for a finite alysidal set Aal, there is a bijection between the set of possible total orderings of the alysidal elements and the set of coupling bijections from Aal to Aal. That is to say, the number of permutations of elements of Aal is the same as the number of total orderings of that set. 4
1.2. Sheaf theory We suppose now that they are not clockwise freeways but sheaves going from the alysidal set Aal to the alysidal set Bal. All the conditions previously expressed for coupling functions will be fulfilled, with the exception of the nonexistence of the inverse function. It is because the sheaf goes in a single direction and a reversibility possibility does not exist. 2. ALYSIDAL STRUCTURES The structures of Alysidal Algebra are based on Birkhoff and Mac Lane (1996), Blyth (2005). Let Aal be an alysidal set whose elements are called chains, and that we will denote by ℘kn ,℘lm , etc. Aal contains all the possible chains between relative beings
es
x1, , x 2 ,..., x n . We denote by ⊕ the law by alysidal sum or operation to add chains.
va @
ua .
2.1. Abelian alysidal group Definition 3: ( Aal ,⊕ ) is an Abelian alysidal group if it satisfies the abelian group axioms: Closure: ∀ρ nk , ρ ml ∈ Aal , ρ nk ⊕ ρ ml ∈ Aal ,
.s el
Associativity: ∀ρ nk , ρ ml , ρ oj ∈ Aal , (ρ nk ⊕ ρ ml ) ⊕ ρ oj = ρ nk ⊕ (ρ ml ⊕ ρ oj ) Identity element: ∃θ ∈ Aal / ∀ρ iw ∈ Aal , θ ⊕ ρ iw = ρ iw ⊕ θ = ρ iw .
su e
Inverse element: ∀ρ iw ∈ Aal , ∃(− ρ iw ) / ρ iw ⊕ (− ρ iw ) = (− ρ iw ) ⊕ ρ iw = θ
jo
Commutativity ∀ρ nk , ρ ml ∈ Aal , ρ nk ⊕ ρ ml = ρ ml ⊕ ρ nk
au th
or :
Note 1: The neutral element θ means the nonexistence chain. Note 2: The operation ℘kn ⊕ (−℘kn ) means the elimination of a chain.
po C stor p re rin sp t on di ng
2.2. Alysidal ring Definition 4: An alysidal ring is an Abelian alysidal group ( Aal ,⊕ ) , together with a
second binary operation ⊥ , such that ∀ρ nk , ρ ml , ρ oj ∈ Aal , ( Aal ,⊕, ⊥ ) is required to be a monoid under the operation ⊥ : 1) Closure under ⊥ : ∀ρ nk , ρ ml ∈ Aal , ρ nk ⊥ ρ ml ∈ Aal 2) ρ nk ⊥ (ρ ml ⊥ ρ oj ) = (ρ nk ⊥ ρ ml ) ⊥ ρ oj
3) Distributive property: ρ nk ⊥ (ρ ml ⊕ ρ oj ) = (ρ nk ⊥ ρ ml ) ⊕ (ρ nk ⊥ ρ oj )
4) Distributive property: (ρ nk ⊕ ρ ml ) ⊥ ρ oj = (ρ nk ⊥ ρ oj ) ⊕ (ρ ml ⊕ ρ oj ) Note 3: The operation ⊥ means selection or possibility of chain. An alysidal ring is a ring with unity, because there exists a multiplicative identity in the alysidal ring, that is, an element θ such that for ∀ρ iw ∈ Aal
ρ iw ⊥ θ = θ ⊥ ρ iw = ρ iw Note 4: The multiplicative identity θ is the same as the neutral element of an abelian alysidal group.
5
An alysidal ring is a commutative ring, because ρ nk ⊥ ρ ml = ρ ml ⊥ ρ nk . Inverse elements for ⊥ in Aal does not exists. Definition 5: An alysidal subset Bal of the alysidal ring ( Aal ,⊕, ⊥ ) which remains a ring when ⊕ and ⊥ are restricted to Bal and contains the multiplicative identity θ of Aal is called an alysidal subring of Aal . Definition 6: For an alysidal ring ( Aal ,⊕, ⊥ ) an alysidal ideal I can be defined as follows:
es
1) ∀ρ nk , ρ ml ∈ Aal , ρ nk ⊕ ρ ml ∈ Aal .
va @
ua .
2) ∀ρ nk , ρ ml ∈ Aal , ((ρ nk ⊥ ρ ml ) ∧ (ρ ml ⊥ ρ nk )) ∈ Aal
.s el
Let ( Aal ,⊕, ⊥ ) be an alysidal ring, with ( Aal ,⊕ ) the underlying additive alysidal group of the alysidal ring.
su e
Definition 7: An alysidal subset Ral of Aal is called alysidal right ideal of Aal if (Ral ,⊕ )
jo
is an alysidal subgroup of ( Aal ,⊕ ) and ∀ρ nk ∈ Ral , ∀ρ ml ∈ Aal , ρ nk ⊥ ρ ml ∈ Aal .
or :
Definition 8: An alysidal subset Lal of Aal is called alyisidal left ideal of Aal if (Lal ,⊕ )
au th
is an alysidal subgroup of ( Aal ,⊕ ) and ∀ρ nk ∈ Lal , ∀ρ ml ∈ Aal , ρ nk ⊥ ρ ml ∈ Aal .
po C stor p re rin sp t on di ng
Let Aal , Bal be two alysidal rings.
Definition 9: An alysidal ring homomorphism is a function f alnxm : Aal → Bal such that ∀ρ nk , ρ ml ∈ Aal
(
)
( )
( )
1) f alnxm ρ nk ⊕ ρ ml = f alnxm ρ nk ⊕ f alnxm ρ ml
2) f alnxm (ρ nk ⊥ ρ ml ) = f alnxm (ρ nk ) ⊥ f alnxm (ρ ml ) 3) f alnxm (θ ) = θ 4) f alnxm (− (ρ nk )) = − f alnxm (ρ nk )
Definition
(
) {
10:
We
define
as
( ) }
alysidal
kernel
of f alnxm ,
ker f alnxm = ρ iw ∈ Aal : f alnxm ρ iw = θ an alysidal ideal in Aal .
Definition 11: The alysidal homomorphism
(
)
ker f alnxm = {θ }.
6
f alnxm
is alysidally injective if
Definition 12: The alysidal image of f alnxm , ima ( f alnxm ) , is an alysidal subring of Bal. Definition 13: If f alnxm is alysidally bijective, then its inverse
(f )
nxm −1 al
is also a ring
alysidal homomorphism and f alnxm is called an alysidal isomorphism, and the rings Aal and Bal are called alysidally isomorphic.
es
Being θ the alysidal ring’s multiplicative element and θ the additive identity element the alysidal characteristic of an alysidal ring Aal , denoted char( Aal ), is defined to be the smallest number of times one must add the ring's multiplicative identity element ( θ ) to itself to get the additive identity element ( θ );that is, char( Aal ) is the smallest positive integer n such that θ ⊕ θ ⊕ ... ⊕ θ = θ . In this case θ ⊕ θ = θ , then char( Aal ) = 1.
va @
ua .
If f alnxm : Aal → Bal then the alysidal characteristic of Bal, char (Bal ) = 1 divides the alysidal characteristic of Aal , char( Aal ) = 1 .
.s el
If Aals is the smallest alysidal subring contained in Aal and Bals is the smallest alysidal subring contained in Bal , then every ring alysidal homomorphism
su e
f alnxm : Aal → Bal induces a ring alysidal homomorphism s f alnxm : Aals → Bals .
jo
2.3. Alysidal left R-module over the alysidal ring
and an operation
R • Aal → Aal called scalar alysidal multiplication,
au th
( Aal ,⊕ )
or :
A left R-module over the alysidal ring ( Aal ,⊕, ⊥ ) consists of an abelian alysidal group ∀r ∈ R, ∀ρ iw ∈ Aal such that ∀r , s ∈ R, ∀ρ nk , ρ ml ∈ Aal , we have
)
po C stor p re rin sp t on di ng
(
r • ρ nk ⊕ ρ ml = r • ρ nk ⊕ r • ρ ml .
(r + s)x = rx + sx (r ⊕ s ) • ρ nk = r • ρ nk ⊕ s • ρ nk .
(rs)x = r(sx) (r • s ) • ρ nk = r • (s • ρ nk ) 1R • ρ nk = ρ nk if R has identity 1R .
Note 5: Scalars can be magnitudes such as intensity of a relation, etc. Note 6: Alysidal sum of scalars: ∀λ , µ ∈ k ., λ ⊕ µ = λ + µ . Suppose Aalmod is a left R-module and Balmod is a subgroup of Aalmod . Definition 14: Balmod is an alysidal R-submodule if, for any n in N and any r in R, the product r • ρ iw ∈ Balmod .
7
We suppose Aalmod and Balmod are alysidal left R-modules. Definition 15: The map f alnxm : Aalmod → Balmod is a homomorphism of alysidal Rmodules if, for any m, n in M and r, s in R ρ nk , ρ ml ∈ Aalmod and r , s ∈ R ,
(
)
( )
( )
f alnxm r • ρ nk ⊕ s • ρ ml = r • f alnxm ρ nk ⊕ s • f alnxm ρ ml
Definition 16: We define as alysidal kernel of an alysidal module homomorphism f alnxm : Aalmod → Balmod the alysidal submodule of Aalmod consisting of all elements that are
es
sent to zero by f alnxm .
.s el
va @
ua .
The well-known isomorphism theorems are valid for alysidal R-modules. (see McLarty, 2006). Let Aalmod and Balmod be alysidal modules, and let g almxn : Aalmod → Balmod be a homomorphism. Then:
su e
1) The kernel of g almxn is an alysidal submodule of Aalmod .
jo
2) The image of g almxn is an alysidal submodule of Balmod .
(
or :
3) The image of g almxn is isomorphic to the quotient alysidal module M / ker(φ)
)
au th
Aalmod / ker g almxn .
po C stor p re rin sp t on di ng
Let Aalmod be an alysidal module, and let X almod and Yalmod be submodules of Aalmod . Then: 1)
The
{
sum X almod ⊕ Yalmod = ρ nk ⊕ ρ ml ρ nk ∈ X almod , ρ ml ∈ X lmod
}
is
an
alysidal
submodule of Aalmod .
2) The intersection X almod Yalmod is an alysidal submodule of Aalmod . mod mod 3) The quotient modules (X al ⊕ Yal )
mod al
Y
mod and (X al )
(X
mod al
Yalmod
) are isomorphic.
Let Aalmod be an alysidal module, and let X almod and Yalmod be alysidal submodules of Aalmod with Yalmod ⊆ X almod ⊆ Aalmod . Then mod 1) The quotient X al
is an alysidal submodule of the quotient M / T mod
Yal
8
Aalmod
Yalmod
.
2) The quotient
Aalmod mod Y al
X almod mod Y al
is isomorphic to
Aalmod
X almod
.
The alysidal left R-modules, together with their module homomorphisms, form an abelian alysidal category. 2.4. The alysidal space Definition 17: Aal is an alysidal space on a body of scalars if is verified that:
(ρ
Asociative k n
⊕ρ
l m
)⊕ ρ
(
∀℘kn ,℘lm ,℘oj ∈
property:
va @
b)
ua .
es
1) If there exists a law of internal composition in Aal, for which Aal, has the structure of an abelian group, we denote as ⊕ the law by alysidal sum and with a neutral element by θ , having, therefore, to verify: a) Internal law: ∀ ℘kn ,℘lm ∈ Aal , ρ nk ⊕ ρ ml ∈ Aal
)
=ρ ⊕ ρ ⊕ρ .
j o
k n
l m
j o
Aal,
.s el
c) Conmutative property: ∀℘kn ,℘lm ∈ Aal , ρ nk ⊕ ρ ml = ρ ml ⊕ ρ nk .
su e
d) Neutral element: ∀℘kn ∈ Aal, ℘kn ⊕ θ = θ ⊕ ℘kn =℘kn .
jo
e) Opposite element: ∀℘kn ∈ Aal , ∃(−℘kn ) so that ℘kn ⊕ (−℘kn ) = θ .
au th
or :
2) There exists on Aal a law of external composition and we denote as ( • ), whose domain of operators is a body of scalars k with the following properties ( ∀λ , µ ∈ , ∀℘kn ,℘lm ∈ Aal ):
(
)
a) External law: ∀ λ∈ y ∀℘kn ∈ Aal , λ • ρ nk ∈ Aal .
po C stor p re rin sp t on di ng
b) Distributive property with respect to a sum of scalars: (λ ⊕ µ) • ℘kn = λ • ℘kn ⊕ µ • ℘kn .
(
)
c) Distributive property with respect to a sum of vectors: λ • ρ nk ⊕ ρ ml = λ•℘ ⊕ λ•℘ . k n
l m
d) Asociative property respect to scalars: λ • (µ • ℘kn ) = (λ • µ) • ℘kn .
e) 1 • ℘kn = ℘kn . An alysidal space has the following immediate properties: 1) ∀℘kn ∈ Aal, 0 • ℘kn = 0. 2) ∀λ ∈ , λ • 0 = 0. 3) If λ • ρ nk = 0 then λ = 0 or ℘kn = 0. 4) ∀λ ∈ , ∀℘kn ∈ Aal, (−λ) • ℘kn = − λ • ρ nk = λ • (−℘kn ). Definition 18: An alysidal subspace of Aal , is a nonempty subset Hal of the alysidal space Aal that has a structure of an alysidal space with the same laws of Aal . For Hal is to be an alysidal subspace of Aal , it is necessary and sufficient that the following conditions are satisfied: 9
i) ∀ ℘kn ,℘lm ∈ Hal, (ρ nk ⊕ ρ ml ) ∈ Hal
ii) ∀α ∈ y ∀℘kn ∈ Hal , (α • ρ nk ) ∈ Hal Definition 19: We define as alysidal intersection of alysidal subspaces Hal and Gal, of alysidal space Aal , and we denote as Hal Gal,, to the alysidal subspace:
{
H al Gal = ℘kn ∈ Aal ℘kn ∈ H al ∧℘kn ∈ Gal
}
es
Definition 20: We define as alysidal sum of alysidal subspaces Hal and Gal, of alysidal space Aal, and we denote as Hal (+) Gal,, to the alysidal subspace: H al (+)Gal = {ρ nk ∈ Aal / ρ nk = ρ ml ⊕ ρ oj , ρ ml ∈ H al ∧ ρ oj ∈ Gal } nxm al
va @
nxm
ua .
Let Aal, Bal be two alysidal spaces on the same body k and f an alysidal map of Aal on Bal. We say that f is a linear alysidal map or alysidal homomorphism if is verified: al
f ( ρ nk ) ⊕ f ( ρ ml ) , ∀ρ nk , ρ ml ∈ Aal
nxm
nxm
al
al
al
.s el
f ( ρ nk ⊕ ρ ml= )
nxm
al
al
jo
nxm
su e
f ( λ • ρ nk ) = λ • f ( ρ nk ) , ∀ρ nk ∈ Aal , ∀λ ∈ k
nxm
The main properties of alysidal homomorphisms are the following ones:
or :
f (0 ) = 0
al
au th
nxm
a)
b) ∀ρ nk ∈ Aal , f (− ρ nk ) = − f (ρ nk ) nxm
nxm
al
al
po C stor p re rin sp t on di ng
c) The image of Hal of the alysidal space Aal is an alysidal subspace of Bal. nxm d) If Gal is a subspace of alysidal space Bal, then f al Aal .
−1
(Gal ) is an alysidal subspace of
nxm
Let
f : Aal → Bal be an alysidal homomorphism.
al
nxm
Definition 21: We define as alysidal kernel of an alysidal homomorphism f , and we al
denote as Ker f , the set al nxm
( )
Ker f = ρ nk ∈ Aal / f ρ nk = 0 al al nxm
nxm
Property 1: The alysidal kernel is an alysidal subspace of Aal. nxm
Definition 22: We define as alysidal image of an alysidal homomorphism
f , and we al
nxm denote as Im f , to the subset of Bal formed by the chains ℘lm that they are images of al some of Aal,
10
{
nxm nxm ρ ml ∈ Bal / ρ ml = f ( ρ nk ) , ρ nk ∈ Aal Im f = al al
}
Property 2: The image of an alysidal homomorphism is an alysidal subspace of Bal. 2.5. Linear combination and composition of alysidal homomorphisms nxm
nxm
al
al
Let f : Aal → Bal and g : Aal → Bal be two alysidal homomorphisms.
n1 xm1
n2 xm2
al
al
va @
g : Bal → C al be two alysidal homomorphisms.
.s el
Let f : Aal → Bal and
ua .
es
Property 3: The linear combination of alysidal homomorphisms is an alysidal homomorphism, that is: nxm nxm The alysidal homomorphism λ ⋅ f ⊕ µ ⋅ g : Aal → Bal , λ , µ ∈ k is an alysidal al al homomorphism.
or :
jo
su e
Property 4: The composition of alysidal homomorphism is an alysidal homomorphism, that is: n2 xm2 n1xm1 The alysidal homomorphism g f : Aal → C al is an alysidal homomorphism. al al
au th
3. THE GNORPSIC FUNCTION It has been specified in previous paragraphs that, if an alysidal element ℘ik of Aal has n
po C stor p re rin sp t on di ng
nodes and the alysidal element ℘lj of Bal has m nodes, then the space of possibilities of coupling will be nxm. Nevertheless, in this space of possibilities, a single one "is chosen" as much by an alysidal element ℘ik as by the ℘lj . Other possibilities are rejected, how if alysidal element ℘lj ∈ B al "knew" in that certain node must make coupling. Therefore, we will have to define a function of knowledge or gnorpsic (of the Greek γνωρψία: to know) associated to the connection between alysidal element ℘ik ∈ Aal and the ℘lj ∈ B al . mj
Definition 23: We define as gnorpsic function and we denote as ω
f℘
k i
→ ℘lj the
ni
function that determines that node ni (departure node) of alysidal element ℘ik ∈ Aal is connected with node mj (arrival node) of alysidal element ℘lj ∈ B al . If the connection of ni (departure node) is only with a single arrival node mj, the mj
function will be mononorsic and we denote as 1 f ℘ik → ℘lj . If the connection of ni ni
(departure node) is with two arrival nodes mj, mk, the function will be bignorsic and we
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m j , mk
denote as 2 f ℘ik → ℘lj . If the connection of ni (departure node) is only with three ni
m j , m k , ml
arrival node mj, mk, ml the function will be trinorsic and we denote as
3
℘ik → ℘lj .
f
ni
If the connection of ni (departure node) is with many arrival node mj, mk, ml, …,mω the m j , mk ,..., mω
function will be polinorsic and we denote as
ω
f
℘ik → ℘lj .
ni
va @
ua .
es
Subindex ni indicates the departure node, supraindex mj the arrival node and supraindex ω the order of coupling. i. Each pair of connected alysidal elements will have, therefore, one gnorpsic associate function. ii. Gnorpsic function will depend on modal components such as necessity, obligation, permission and faculty.
.s el
4. APPLICATION OF ALYSIDAL ALGEBRA: THEORY OF IDEOLOGIES In figure 1 we represented the hypothesis of the relationship between structural base (SB)-doxical and mythical superstructures (Usó-Domènech et al, 2009a,b).
su e
Mythical Superstructure (MS)
1
IDEAL
Primigenial Base (PB)
2
Ideal Structure (ISt)
jo
Ideal Values, Myths.
Ideal V alues, abstract ideology Utopia (Goals)
or :
3
au th
inverse-MS-image mythical superstructural image (MS-image) 3
last goal inverse-MS-projection (concretion)
po C stor p re rin sp t on di ng
Doxical Superstructure (DS)
V alues in f act, Dominant Ideology, Culture: Science, A rt, Folk belief s, etc.
doxical superstructural image (DS-image).
Subject
connotative-SB- projection (action) near goal
4 1
2
Actual Structural Base
Desirable Structural Base
ACTUAL
Figure 1 The Ideal will be formed by the Mythical Superstructure MS = (PB ISt ) and the present thing formed by the Doxical Superstructure (DS) and the Structural Base (SB). An abstract ideal ideological system is not identical with the concrete actual description of ideology (DS) and behavior (SB). Witness the uncomfortable interactions between
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the ideal of democratic politics and the operation of a national convention or the interaction between Christian theology and the operation of a church bureaucracy. On other hand, humans recognize the distance between the ideal and the actual. In so doing, humans apply two different sets of standards and two different conceptions of cause and effect. Definition 24: We define as belief ideal distance di the abstract distance in the mind of the believer (or believers) existing between the ideal and the actual, that is to say, between which it is desired and what is.
.s el
va @
ua .
es
Beliefs distances are not equal. It does not exist the same belief ideal distance between abstract ideal and concrete ideologies that the existing belief ideal distance between ideal values in and values fact and the long ago greater between immediate and last goals. If we represented a three-dimensional space both superstructures and we suppose that the Doxical Superstructure forms a plane, the ISt-Mythical Superstructure would form a warped plane (Figure 2) based on the belief ideal distances, plane that will change its form in as much the distances are modified approaching or moving away the present thing to the ideal.
su e
1
(IDEAL)
po C stor p re rin sp t on di ng
au th
or :
jo
ISt-Mythical Superstructure
d1
d2
d3
......... d n
2
Doxical Superstructure
(ACTUAL)
Figure 2 The abstract ideal ideology should be parallel, point for point, to one or more concrete expresions of the ideology. Thus the abstract ideal ideology is often used as a practical excuse for a parallel system of concrete beliefs (concrete ideology) that is quite different. The operation of ideologies in social behavior through the denotative-SBprojection, involve orientation to both abstract and concrete sets of goals and values at the same time.
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We consider the Ist-Mythical Superstructure and The Doxical Superstructure (DS) like two alysidal sets with unequal cardinal number of alysidal elements. We call Aal and Bal to two alysidal sets (ISt-Mythical Superstructure and Doxical Superstructure). By simplification, we suppose that Aal has three alysidal elements (ρ n11 , ρ n22 , ρ n33 ) corresponding to Ideal Abstract Ideology, Ideal Values and Utopia respectively. Bal has w alysidal w elements (ρ n11 , ρ n22 , ρ n33 ....ρ nw ) corresponding Concrete ideology, Values in fact, Inmediate goals, Art, Science, Folk beliefs, Religion and so on. Subscripts n1 , n 2 , n 3 , m1 , m 2 , m 3 ,..., m w . correspond to substantive beliefs considered like nodes in a the theory of Alysidal sets. The coupling function f : ( Aal ) = B al is an injective coupling function. In Alysidal Sets Theory (AST), each alysidal element is a chain formed by n interrelated elements (nodes). In this case, the nodes are substantive beliefs interrelated with abstract relations. We consider the Abstract Ideology belonging to ISt-Mythical Superstructure and the Dominant Ideology belonging to Doxical Superstructure like two alysidal sets with an only element (Figure 3)
va @
ua .
es
3 Xw
.s el
ISt-Mythical Superstructure
(AI)
5
2
3
departure nodes
jo
1
su e
Abstract Ideal Ideology
6
po C stor p re rin sp t on di ng
au th
or :
4
Concretion gnorpsic functions
arrival nodes i
f
b
j g
a
k
e
d
c
Concrete Ideology
l h
(CI)
ideological Doxical Superstructure
Figure 3 Nodes are substantive beliefs and binary relations are abstract relations between substantive beliefs. In figure 3 the following pairs have formed: (1, a), (2, b), (3, d), (4, e), (5, f), (6, g). Abstract ideology will be the domain and concrete ideology will be the codomain. In AST if one alysidal element ℘ik of Aal has n nodes and the alysidal element ℘lj of Bal has m nodes, the space of possibilities of coupling will be nxm. Nevertheless, in this
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space of possibilities, a single one "is chosen" so much by alysidal element ℘ik as by the
℘lj . The other possibilities are rejected, how if alysidal element ℘lj ∈ B al "knew" in that certain node must make coupling. Therefore, we will have to define a function of knowledge or gnorpsic (of the Greek γνωρψία: to know) associated to the connection between alysidal element ℘ik ∈ Aal and the ℘lj ∈ B al . The gnorpsic function mj
ω
f℘
k i
→ ℘lj is the function that determines that node ni (departure node) of alysidal
ni
element ℘ik ∈ Aal is connected with node mj (arrival node) of the alysidal element
es
℘lj ∈ B al . Subindex ni indicates the departure node, supraindex mj the arrival node and supraindex ω the order of coupling. mj
ni
ua .
In the case that occupies to us it is a mononorpsic function 1 f ℘ik → ℘lj ,but with a
va @
special meaning: the concrection of the ideals with concrete substantive beliefs. Let AI and CI be the abstract ideal and concrete ideologies so that AI ∈ Aal ∧ CI ∈ B al .
.s el
mj
f
( AI ) con → CI the
ni
su e
Definition 25: We define as concretion function and we denote as
jo
monognorpsic function that determines that substantive ideal belief ni (departure substantive ideal belief) of alysidal element AI ∈ ISt − Mythical Superstructure is
au th
or :
connected with the concrete substantive belief mj (arrival concrete substantive belief) of the alysidal element CI ∈ DS . In the case of figure 3, the concretion functions are: a
( AI ) con → CI
po C stor p re rin sp t on di ng
f 1
b
f
( AI ) con → CI
2
d
f
( AI ) con → CI
3 e
f
( AI ) con → CI
4 f
f
( AI ) con → CI
5 g
f
( AI ) con → CI
6
These concretion functions turn ideal substantive beliefs in concrete substantive beliefs. An ideal legal structure (ideal deontical normative) becomes one concrete legal structure (makes specific and concrete deontical normative) and it is projected on Structural Base determining the behavior of the society. Subjectivity, prevarication, parciality, etc of the legislative body comes certain in last instance by the double ideological content: ideal and concrete.
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REFERENCES Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. Macmillian. New York. 1996. Blyth, T.S. Lattices and Ordered Algebraic Structures. Springer.2005. McLarty, C. Emmy Noether’s Set Theoretic Topology: From Dedekind to the rise of functors. In The Architecture of Modern Mathematics: Essays in history and philosophy Edited by Jeremy Gray and José Ferreirós. Oxford University Press. 2006.
va @
ua .
es
Usó-Domènech, J.L., Vives-Maciá, F., Nescolarde-Selva, J. and Patten, B.C. A Walford’s metadynamic point of view of ecosustainability ideology (I). In: Sustainable Development and Global Community. Vol X. Edited by George E. Lasker and Kensei Hiwaki. The International Institute for Advanced Studies in Systems Research and Cybernetics. 35-39. 2009a.
po C stor p re rin sp t on di ng
au th
or :
jo
su e
.s el
Usó-Domènech, J.L., Vives-Maciá, F., Nescolarde-Selva, J. and Patten, B.C. A Walford’s metadynamic point of view of ecosustainability ideology (II). In: Sustainable Development and Global Community. Vol X. Edited by George E. Lasker and Kensei Hiwaki. The International Institute for Advanced Studies in Systems Research and Cybernetics. 41-45. 2009b.
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