,
Mircea Sularia
"Politehnica" University of Bucharest Department of Mathematics II 313 Splaiul Independentei, 77206 Bucharest 5 ROMANIA e-mail: sularia@euler.math.pub.ro
Abstract: The variety of biresiduated algebras including the structures of D-algebra and MV-algebra was introduced together with a corresponding logical system. A many-valued space over a biresiduated algebra is a set equipped with an equivalence function and a distance function such that these functions are complementary. A cartesian closed category of many-valued spaces is presented.
Keywords: fuzzy set, MV-algebra, Heyting algebra,
Brouwer algebra, D-algebra, residuated lattice, biresiduated algebra, equivalence
function, distance function, category.
Introduction
The notion of fuzzy set was introduced by L. A. Zadeh in 1965 as "a class of objects with a continuum of grades of membership" [12]. A fuzzy set A is characterized by a mapping fA from X to [0, 1], called membership function on X, where [0, 1] ÌR is the complete bounded chain of positive real numbers. In this paper the acceptance of a fuzzy set is that of a couple (X, f), where X is a set and f : X ® [0, 1] is a function.
The notion of L-set including the notion of fuzzy set was introduced by J. A. Goguen as a couple (X, f), where L is a lattice and f : X ® L is a function. Goguen considers that the algebra of inexact concepts is a residuated lattice [6]. Adjoint couples and residuated lattices are often used in the fuzzy set theory [11].
The theory of MV-algebras is a mathematical development arising from algebraic foundations of many-valued reasoning [3, 4].
An MV-algebra has both a structure of residuated lattice and a structure of dual residuated lattice.
In order to identify a standard logical system which includes features common to some basic many-valued logical systems, a variety of biresiduated algebras was introduced in [10]. This class of biresiduated algebras includes Heyting algebras [1, 2], Brouwer algebras and MV-algebras. A D-algebra [9, 10] is a structure isomorphic to a subdirect product between a Heyting algebra and a Brouwer algebra. Thus, every D-algebra is also a biresiduated algebra.
A general description of the connection between some basic algebraic structures from the category of biresiduated algebras, is given.
The notions of equivalence and distance functions on a set over a biresiduated algebra are introduced together with the notion of many-valued space. Different examples of these notions are given. The purpose of this paper is to present a cartesian closed category of many-valued spaces over a complete D-algebra. This category can be considered as a starting point leading to a new suitable mathematical development of the fuzzy set theory.
1. Basic algebraic structures
1.1 Biresiduated algebras
Let K be the class of algebras
A = (A, Ù ,
Ú
, Ä , ®
, Å , - , Ø
, 0, 1)
of type (2, 2, 2, 2, 2, 2, 1, 0, 0).
A biresiduated algebra is an algebra A of K with seven operations Ù (meet), Ú (join), Ä (multiplication), ® (residuation), Å (addition), - (dual residuation), Ø (negation) and two constants 0, 1 Î A such that:
(BR1) (A, Ù , Ú
, 0, 1) is a bounded distributive lattice with the minimum element 0
and the maximum
element 1.
(BR2) (A, Ä ) and (A,
Å
) are commutative semigroups.
(BR3) The following equations hold:
(i) x Ú (x Ä
y) = x
(io) x Ù
(x Å y) = x
(ii) x Ä (y Ú
z) = (x Ä y) Ú
(x Ä z)
(iio) x Å
(y Ù z) = (x Å
y) Ù (x Å
z)
(iii) x Ä (y Ù
z) = [(x Ä y) Ù
(x Ä z)] Ä
1
(iiio) x Å
(y Ú z) = [(x Å
y) Ú (x Å
z)] Å 0
(iv) (x Ä y) Ä
1 = x Ä y
(ivo) (x Å
y) Å 0 = x Å
y
(v) (x Ä y) Å
0 = (x Å 0) Ä
(y Å 0)
(vo) (x Å
y) Ä 1 = (x Ä
1) Å (y Ä
1)
(vi) (x ® y) Ä
1 = x ® y;
(vio) (x - y) Å
0 = x - y;
( vii) (x ® y) Å
0 = Ø (x -
y)
(viio) (x - y)
Ä
1 = Ø (x ®
y)
(viii) Ø x = x ®
0
(viiio) Ø
x = 1 - x
(ix) x ® (y ®
z) = (x Ä y) ®
z
(ixo) (x - y) -
z = x - (y Å
z)
(x) x Ä (x ®
y) = (x Ù y) Ä
1
(xo) (x - y) Å
y = (x Ú y) Å
0
(xi) (x Ù y) ®
x = 1
(xio) x - (x Ú
y) = 0
(xii) (x Ä 1) ÚØ
x = x ÚØ x
(xiio) (x Å
0) Ù Ø
x = x Ù Ø
x
Let BR be the class of biresiduated algebras.
1.2 Residuated algebras
A biresiduated algebra will be called residuated algebra if verifying:
(R1) x Ä 1 = x.The following condition holds in every residuated algebra:
z £ x ® y iff z Ä x £ y.Let R be the class of residuated algebras.
1.3 Heyting algebras
A Heyting algebra is a system
(A, Ù , Ú , ® , 0, 1)such that (A, Ù , Ú , 0) is a relatively pseudo-complemented lattice with the minimum element 0, the binary operation of relative pseudocomplementation ® and 0, 1 Î A satisfying 1 = 0 ® 0 and for every x, y, z Î A:
z £ x ® y iff z Ù x £ y.Let H be the class of Heyting algebras and H* be the class of biresiduated algebras verifying the following equation:
(R2) x Ä y = x Ù y.The next result expresses the relation between the classes of algebras H*, R, BR and H.
1.4 Proposition
(i) H* ÌRÌBR.
(ii) The algebraic categories associated with the classes H and H* are isomorphic.
Proof
( i) Relation 1.3(R2) implies 1.2(R1).
(ii) A biresiduated algebra of H* can be associated with every Heyting algebra
(A, Ù , Ú , ® , 0, 1) ÎHsuch that it verifies 1.3(R2) together with the following equations:
Ø x = x ® 0;Using this correspondence an isomorphism between the algebraic categories associated with H and H* is reached. ÿ
x - y = Ø (x ® y)
x Å y = ØØ (x Ú y).
1.2o Dual residuated algebras
A biresiduated algebra will be called dual residuated algebra if verifying:
(R1o) x Å 0 = x.The following condition holds in every dual residuated algebra:
x - y £ z iff x £ y Å z.Let Ro be the class of dual residuated algebras.
1.3o Brouwer algebras
A Brouwer algebra is a system
(A, Ù , Ú , - , 0, 1)such that (A, Ù , Ú , 1) is a dual Heyting lattice with a binary operation - (relative pseudo-subtraction) and 0, 1 Î A satisfying 1 = 0 - 0 and for every x, y, z Î A:
x - y £ z iff x £ y Ú z.Let Br be the class of Brouwer algebras and Br* be the class of biresiduated algebras verifying the following equation:
(R2o) x Å y = x Ú y.The next result expresses the relation between the classes of algebras Br*, Ro, BR and Br.
1.4o Proposition
(i) Br* ÌRoÌBR.
(ii) The algebraic categories associated with the classes
Br
and Br* are isomorphic.
Proof
( i) Relation 1.3o(R2o) implies
1.2o(R1o).
(ii) A biresiduated algebra of Br*
can be associated with every Brouwer algebra
(A,^,Ú ,-,0,1)ÎBrsuch that it verifies 1.3o(R2o) together with the following equations:
Ø x = 1 - x;Using this correspondence an isomorphism between the algebraic categories associated with Br and Br* is obtained. ÿ
x ® y = Ø (x - y);
x Ä y = ØØ (x Ù y).
1.5 MV-algebras
An MV-algebra [4] is a system (A, Å , Ø , 0) of type (2, 1, 0) such that the following equations hold:
MV1) x Å (y Å z) = (x Å y) Å z;From MV1)-MV3) it follows that (A, Å , 0) is an abelian monoid. We define a constant 1 and the operations Ä , ® and - together with a binary relation £ on A as follows, for any two elements x, y of A:
MV2) x Å y = y Å x;
MV3) x Å 0 = x;
MV4) ØØ x = x;
MV5) x Å Ø 0 = Ø 0;
MV6) Ø (Ø x Å y) Å y = Ø (Ø y Å x) Å x.
( 1) 1 = Ø 0;Then £ is an order relation which determines on A a structure of distributive lattice with the smallest element 0 and the greatest element 1, (A, Ù , Ú , 0, 1), such that:
( 2) x Ä y = Ø (Ø x ÅØ y);
( 3) x ® y = Ø x Å y;
( 4) x - y = x ÄØ y;
( 5) x £ y iff Ø x Å y = 1.
( 6) x Ú y = (x Ä Ø y) Å y;Let MV be the class of MV-algebras and MV* be the class of algebras
( 7) x Ù y = Ø (Ø x ÚØ y).
A* = (A, Ù , Ú , Ä , ® , Å , - , Ø , 0, 1)associated with an MV-algebra
A = (A, Å , Ø , 0),where the operations Ù , Ú , Ä , ® , Ø and 1 are defined as above. The next result expresses the relation between the classes MV, MV* and BR.
1.6 Proposition
(i) MV* = R ÇRoÌBR.
(ii) The algebraic categories associated with the classes
MV
and MV* are isomorphic.
Proof
( i) Every algebra A* ÎMV* associated with an MV-algebra A ÎMV and defined as in 1.5 is a biresiduated algebra (see Definition 1.1) satisfying equations 1.2(R1) and 1.2o(R1o). The following specific relations hold:
x Ù y = x Ä (x ® y);(ii) Using the precedent correspondence an isomorphism between the algebraic categories associated with MV and MV* can be obtained.ÿ
x Ú y = (x - y) Å y;
1.7 Theorem
The class BR of biresiduated algebras is the variety of algebras of K generated by R ÈRo.
1.8 Boolean algebras
A Boolean algebra [8] is a system
(A, Ù , Ú , Ø , 0, 1)of type (2, 2, 1, 0, 0) such that (A, Ù , Ú , 0, 1) is a bounded distributive lattice and it satisfies the equations x ÙØ x = 0 and x ÚØ x = 1.
Let B be the class of Boolean algebras and B* be the class of algebras A Î K associated with Boolean algebras (A, Ù , Ú , Ø , 0, 1) such that:
x Ä y = x Ù y;1.9 D-algebras
x ® y = Ø x Ú y;
x Å y = x Ú y;
x - y = x ÙØ y.
A D-algebra [9] is a system
A = (A, Ù , Ú , ® , - , 0, 1)of type (2, 2, 2, 2, 0, 0) such that it is isomorphic to a subdirect product of two structures
H = (H, Ù , Ú , ® , - , 0, 1)and
Br = (Br, Ù , Ú , ® , - , 0, 1),where (H, Ù , Ú , ® , 0, 1) is a Heyting algebra such that x - y = (x ® y) ® 0, for all x, y Î H and (Br, Ù , Ú , - , 0, 1) is a Brouwer algebra such that x ® y = 1 - (x - y), for all x, y Î Br.
Let D be the class of D-algebras and D* be the class of biresiduated algebras A ÎBR such that the following equations hold:
(D1) x Ä y = (x Ù y) Ä 1;Then the algebraic categories associated with D and D* are isomorphic.
(D2) x Å y = (x Ú y) Å 0.
The structures of Boolean algebra, Heyting algebra and Brouwer algebra are related to the structure of D-algebra as follows:
1.10 Theorem
( i) B* = H* ÇBr*ÌD*ÌBR.
(ii) D* is a variety of biresiduated
algebras generated by H* ÈBr*.
2. Many-valued spaces over a biresiduated algebra
Let X be a set and A be a biresiduated algebra.
In this Section the notions of equivalence and distance functions on X and the term of many-valued space over A will be introduced.
2.1 Definition
An equivalence function on X over A is a mapping e : X ´ X ® A Ä 1 such that the following conditions hold for all x, y, z Î X:
( i) e(x, x) = 1;2.2 Definition
( ii) e(x, y) = e(y, x);
(iii) e(x, y) Ä e(y, z) £ e(x, z).
A distance function on X over A is a mapping
d : X ´ X ® A Å 0such that the following conditions hold for all x, y, z Î X:
( i) d(x, x) = 0;2.3 Definition
( ii) d(x, y) = d(y, x);
(iii) d(x, z) £ d(x, y) Å d(y, z).
A many-valued space over A with the carrier set X (called more simply an A-valued space) is a system X = (X, e, d) such that
e : X ´ X ® A Ä 1is an equivalence function on X over A,
d : X ´ X ® A Å 0is a distance function on X over A and the following conditions hold for all x, y Î X:
( i) e(x, y) Ä d(x, y) = 0;2.4 Example
(ii) e(x, y) Å d(x, y) = 1.
Let R Í X ´ X be an equivalence relation on X,
eR = aR : X ´ X ®{0, 1}be the Boolean characteristic function of R, i.e.
for all x, y Î X and
dR = ØaR : X ´ X ® {0, 1}be the Boolean complement of aR, i.e.
for all x, y Î X.
Then the standard A-valued space associated with R is the triplet
X[R]= (X, eR, dR).Thus, the standard A-valued space associated with the identity relation R on X,
R = D = {(x, x) / x Î X},is the system
X[D ] = (X,where for all x, y Î X:,
)
and
2.5 Example
Let
eA : A ´ A ® A Ä 1and
dA : A ´ A ® A Å 0be the mappings defined for all x, y Î A by
eA(x, y) = (x ® y) Ä (y ® x);Then the standard A-valued spaceassociated with A is the triplet
dA(x, y) = (x - y) Å (y - x).
A = (A, eA, dA).The mapping eA is called the equivalence function on A and the mapping dA is called the distance function on A.
2.6 Example
If A Î D* is associated with a D-algebra
(A, Ù , Ú , ® , - , 0, 1)then the equivalence and distance functions on A are defined by:
eA(x, y) = [(x ® y) Ù (y ® x)] Ä 1;for all x, y Î A.
dA(x, y) = [(x - y) Ú (y - x)] Å 0,
If A Î H*ÌD* is associated with a Heyting algebra (A, Ù , Ú , ® , 0, 1) then for all x, y Î A:
eA(x, y) = (x ® y) Ù (y ® x);where Ø u = u ® 0, " u Î A. Then concrete expressions of equivalence and distance functions on A can be obtained, if A is associated with the complete Heyting algebra of open subsets of a topological space.
dA(x, y) = ØØ [Ø (x ® y) Ú Ø (y ® x)],
If A Î Br*ÌD* is associated with a Brouwer algebra (A, Ù , Ú , - , 0, 1) then for all x, y Î A:
eA(x, y) = ØØ [Ø (x - y) ÙØ (y - x)];where Ø u = 1 - u, " u Î A. Then concrete expressions of equivalence and distance functions on A can be also obtained, if A is associated with the complete Brouwer algebra of closed subsets of a topological space.
dA(x, y) = (x - y) Ú (y - x),
If A Î B*ÌD* is associated with a Boolean algebra (A, Ù , Ú , Ø , 0, 1) then for all x, y Î A:
eA(x, y) = (Ø x Ú y) Ù (Ø y Ú x);2.7 Example
dA(x, y) = (x ÙØ y) Ú (y ÙØ x).
Let A be the Lukasiewicz structure on the real unit interval [0, 1] i.e. for all x, y Î [0, 1]:
x Ù y = min(x, y);where in the second member of the precedent relations + and - are the usual operations of addition and subtraction of real numbers. Then the following relations hold:
x Ú y = max(x, y);
x Ä y = max(0, x + y - 1);
x ® y = min(1, 1 - x + y);
x Å y = min(1, x + y);
x - y = max(0, x - y);
Ø x = 1 - x,
eA(x, y) = 1 -Therefore, the distance function dA is the usual distance on [0, 1] Ì R and the equivalence function eA is the negation of dA.;
dA(x, y) =
2.8 Example
Let A be a structure of biresiduated algebra on the real unit interval [0, 1] defined by the following relations for all x, y Î [0, 1]:
x Ù y = min(x, y);where in the second member of the precedent relations x × y is the multiplication of x by y and
x Ú y = max(x, y);
x Ä y = x × y;;
;
,
is the division of y by x ¹ 0 in R.
Then the following relations hold:
2.9 Example;
.
Let A be a structure of biresiduated algebra on the real unit interval [0, 1] defined by the following relations for all x, y Î [0, 1]:
x Ù y = min(x, y);The following relations will then hold:
x Ú y = max(x, y);;
;
x Å y = x + y - x × y;;
.
2.10 Example;
.
Let A be the structure of biresiduated algebra on the real unit interval [0, 1] defined as in Example 2.8. Suppose that p Î R and p > 1. Define the mapping
ep : R´R® [0, 1]for all x, y Î R by
ep(x, y) = pIt follows that for all x, y, z ÎR.
ep(x, y) × ep(y, z) £ ep(x, z).Thus ep is an equivalence function on R over A.
Let A be the structure of biresiduated algebra on the real unit interval [0, 1] defined as in Example 2.9. Define the mapping
dp : R´R® [0, 1]for all x, y Î R by
dp(x, y) = 1 - ep(x, y) = 1 - pThen dp is a distance function on R over A. ÿ.
3. The category S[A]
Let A be a biresiduated algebra.
If X, Y, Z are sets and f : X ® Y, g : Y ® Z are functions then g × f : X ® Z is the function defined by (g × f)(x) = g(f(x)), for all x Î X. For all sets X and Y, let [X, Y] be the set of all functions from X to Y. A specific terminology used in fuzzy set theory [7] will be adopted in this section. A function m Î [X, A] is called an A-subset of X. For all f Î [X, Y], if n Î [Y, A] is an A-subset of Y then n × f Î [X, A] is called the inverse image of nunder f.
The next definition introduces a notion of morphism which makes the class S[A] of all A-valued spaces be a category.
3.1 Definition
Let X = (X, eX, dX) and Y = (Y, eY, dY) be two A-valued spaces. A morphism from X toY is a function
f : X ® Ysuch that the following conditions hold for all x1, x2 Î X and y Î Y:
(i) eX(x1, x2) Ä eY(f(x1), y) £ eY(f(x2), y);The set of all morphisms from XtoY will be denoted by Hom(X, Y).
(ii) dY(f(x2), y) £ dY(f(x1), y) Å dX(x1, x2).
3.2 Definition
Let f Î Hom(X, Y) and g ÎHom(Y, Z) be morphisms, where X, Y, ZÎS[A]. Then
g × f ÎHom(X, Z)and it is called the product of the couple (g, f). The class S[A] and the class of all morphisms together with this product of morphisms is a category denoted by S[A].
The next definition introduces the notion of monotone A-subset of an A-valued space and it will be used to characterize morphisms.
3.3 Definition
Let X = (X, e, d) Î S[A]. A monotone A-subset of X is a function
m : X ® Asuch that the following conditions hold for all x, y Î X:
(i) m (x) Ä e(x, y) £ m (y);The set of all monotone A-subsets of X will be denoted by M[X].
(ii) m (y) £m (x) Å d(x, y).
3.4 Lemma
Let X = (X, e, d) Î S[A] and x Î X. Define two functions
e[x] : X ® A and d[x] : X ® Asuch that for all y Î X:
e[x](y) = e(x, y);Then e[x], d[x] Î M[X].
d[x](y) = d(x, y).
Proof
Now we show that e[x] ÎM[X]. The functions e and d satisfy all the conditions 2.1(i)-(iii), 2.2(i)-(iii), 2.3(i) and 2.3(ii). Let x’, y Î X. From Definition 2.1 (ii) it follows that
(1) e[x](x’) Ä e(x’, y) = e(x, x’) Ä e(x’, y)From Definition 2.2 (ii) and the relation
£ e(x, y)
= e[x](y).
(" a Î A ) a Å Ø a = 1it follows that
1 = d(x, x’) Å Ø d(x, x’)This implies that
£ d(x, y) Å d(x’, y) Å Ø d(x, x’).
Ø d(x, y) £Ø d(x, x’) Å d(x’, y),but the conditions 2.3(i) and 2.3(ii) imply
The relations (1) and (2) imply e[x] ÎM[X].
The other condition d[x] Î M[X]
follows using similar arguments. ÿ
The next result characterizes morphisms as functions
preserving the property of monotony by making inverse images.
3.5 Proposition
Let X = (X, eX, dX) and Y = (Y, eY, dY) be two objects of S[A] and f : X ® Y be a function. Then the following conditions are equivalent:
(i) f Î Hom(X, Y);Proof
(ii) n × f ÎM[X], for all n ÎM[Y].
(i) Þ (ii). Suppose that 3.5 (i) holds. Using Definitions 2.3 and 3.1 this implies that for all x1, x2 Î X:
(1) eX(x1, x2) £ eY(f(x2), f(x1);Let n ÎM[Y] i.e. n : Y ® A is a function such that for all y1, y2 Î Y:
(2) dY(f(x2), f(x1)) £ dX(x1, x2).
(3) n (y1) Ä eY(y1, y2) £n (y2);Suppose that x1, x2 Î X. Using (3) and (4) and the properties of the operations Ä and Å in A, from (1) and (2) it follows that:
(4) n (y2) £n (y1) Å dY(y1, y2).
n (f(x1)) Ä
eX(x1, x2) £n
(f(x1)) Ä eY(f(x1),
f(x2))
£ n (f(x2));
n (f(x2)) £n
(f(x1)) Å dY(f(x1),
f(x2))
£ n (f(x1)) Å
dX(x1, x2).
Therefore, n ×
f ÎM[X].
This shows that the condition 3.5 (ii) holds.
(ii) Þ (i). Suppose that 3.5 (ii) holds. Using Lemma 3.4 it follows that for all x1 Î X, the functions
eY[f(x1)] : Y ® Aand
dY[f(x1)] : Y ® Aare monotone A-subsets of Y. From 3.5 (ii) it follows that
eY[f(x1)] × f : X ® Aand
dY[f(x1)] × f : X ® Aare monotone A-subsets of X. This implies that the following relations hold for all x1, x2Î X:
(5) eX(x1, x2) = eX(x1,
x2) Ä 1
= eX(x1, x2) Ä
eY(f(x1), f(x1))
= eX(x1, x2) Ä
(eY[f(x1)] × f)(x1)
£ (eY[f(x1)]
×
f)(x2)
= eY(f(x1), f(x2)).
(6) dY(f(x1), f(x2))
= (dY[f(x1)] × f)(x2)
£ dX(x1, x2) Å
(dY[f(x1)]
× f)(x1)
= dX(x1, x2) Å
dY(f(x1), f(x1))
= dX(x1, x2) Å
0
= dX(x1, x2).
From (5) and (6) it follows that f verifies the conditions from Definition 3.1, i.e. 3.5 (i) holds. This completes the proof. ÿ
3.6 Example
Let 2 be a biresiduated algebra having precisely two elements 0 and 1. Suppose that R Í X ´ X is an equivalence relation on X. Let
X[R] = (X, eR, dR)be the standard 2-valued space associated with R defined as in example 2.4. Then (X, dR) is a metric space such that D Í X is open iff D is R-monotone i.e. the following condition holds for all x, y Î X:
x Î D and (x, y) Î R imply y Î D.Using Definition 3.3, it follows that the following condition holds for all D Í X:
D is R-monotone iff mDÎM[X[R]],where mD : X ® {0, 1} is the characteristic function of D defined for all x Î X by
mDTherefore, the set of all monotone 2-subsets of X[R] can be identified with the set of all open sets of the metric space (X, dR)..
Suppose that
Y[R’] = (Y, eR’, dR’)is another 2-valued space associated with an equivalence relation R’ on Y. From Proposition 3.5 it follows that the following conditions are equivalent for any function f : X ® Y:
(i) f Î Hom(X[R], Y[R’]);Therefore, the category S[A] includes all 2-valued spaces associated with equivalence relations on sets such that the notion of morphism between A-valued spaces is an extension of the notion of continuous functions between standard metric spaces associated with these equivalence relations.
(ii) f is a continuous function from the metric space (X, dR) to the metric space (Y, dR’).
3.7 Example
Let X1 = (X, e1, d1) and X2 = (X, e2, d2) be two A-valued spaces having the same carrier set X and 1X : X ® X be the identity mapping on X. Define X1<X2 iff 1X Î Hom(X1, X2) i.e. the following conditions hold for all x, y Î X:
e1(x, y) £ e2(x, y);Let m : X ® A be any function. Define two functions
d2(x, y) £ d1(x, y).
em , e*m: X ´ X ® A Ä 1such that for all x, y Î X:
em (x, y) = eA(m (x), m (y));where eA is the equivalence function on A defined as in Example 2.5 and
e*m(x, y) = i,
i : A ® A Ä 1is an interior operator on the poset (A, £ ) [2,10] defined by
i(a) = a Ä 1, " a Î A.Define also two functions
dm , d*m: X ´ X ® A Å 0such that for all x, y Î X:
dm (x, y) = dA(m (x), m (y));where dA is the distance function on A defined as in Example 2.5 and
d*m(x, y) = c,
c : A ® A Å 0is a closure operator on the poset (A, £ ) [2,10] defined by
c(a) = a Å 0, " a Î A.Then Xm= (X, em , dm ) and X*m= (X, e*m, d*m) are two A-valued spaces with the same carrier set X such that the following conditions hold:
3.8 Lemma
Let X = (X, eX, dX), Y = (Y, eY, dY) ÎS[A]. Then the following conditions hold in the category S[A] for all f Î Hom(X, Y):
(i) f is a monomorphism iff f is injective.
(ii) If the function f is surjective then f is an epimorphism.
(iii) If the function f is bijective then f is an isomorphism iff for all x1, x2 Î X:
(I1) eX(x1, x2) = eY(f(x1), f(x2));Proof
(I2) dX(x1, x2) = dY(f(x1), f(x2)).
The condition 3.8 (ii) and the fact that if f is an injective function then f is a monomorphism are clear. Suppose that f is a monomorphism, but f is not an injective function i.e. for some x1, x2Î X, we have f (x1) = f(x2) and x1 ¹ x2. Let Z ={ x1, x2 } and g, h : Z ® X such that g(z) = z and h(z) = x1, " z Î Z. This implies that f × g = f × h. Let Z[D ] = (Z, eD , dD ) be the standard structure of A-valued space associated with the identity relation D on Z defined as in example 2.4. Using definition 3.1 one obtains that g, h ÎHom(Z[D ], X). Then, from the property f × g = f × h and the condition that f is a monomorphism it follows that g = h, but the definitions of g and h imply g ¹ h, contradiction. Thus, 3.8 (i) and 3.8(ii) hold. The condition 3.8(iii) follows from the fact that if f is a bijective function then the relations 3.8(I1) and 3.8 (I2) hold iff f-1 ÎHom(Y, X), where f-1 : Y ® X is the inverse function of f.
ÿ
Now we show that the finite direct limits exist in the category S[A]. Concrete constructions of finite products and equalizers are given.
3.9 Finite direct limits
Let I be a nonempty finite set and
(Xi = (Xi, ei, di))i Î Ibe a family of objects in S[A]. Let
X =be the cartesian product set of (Xi)i Î I and
(pi : X ® Xi) i Î Ibe the family of canonical projections of X.
Define two functions
eX : X ´ X ® A Ä 1and
dX : X ´ X ® A Å 0such that for all x, y Î X:
and
Then the triplet.
X = (X, eX, dX)is an object in S[A] such that
( " i Î I ) pi ÎHom(X, Xi).The system (X, (pi) i Î I) is a direct product of the family (Xi)i Î I in the category S[A]. This property is a consequence of the fact that for all Y = (Y, eY, dY) ÎS[A], if
( " i Î I ) pi Î Hom(Y, Xi)then there exists an unique function
f : Y ® Xdefined by
( " y Î Y ) ( " i Î I ) f(y)(i) = pi (y)such that
f Î Hom(Y, X)and
( " i Î I ) pi × f = pi.Let 1[D ] = (1, eD , dD ) be an object of S[A] associated with the identity relation D on a set 1 having precisely one element. Then 1[D ] is a final object for S[A] i.e. the set Hom(X, 1[D ]) has precisely one element, for each XÎS[A]. It follows that the following condition holds:
(2) Equalizers exist in the category S[A].
Suppose that
f, g Î Hom(X, Y),where
X = (X, eX, dX), Y = (Y, eY, dY) ÎS[A].Define a subset Z of X by
Z = { x Î X / f(x) = g(x) }and two functions
eZ : Z ´ Z ® A Ä 1and
dZ : Z ´ Z ® A Å 0such that for all u, v Î Z:
eZ (u, v) = eX (u, v);Let s : Z ® X be the inclusion map of Z in X i.e. ( " z Î Z ) s (z) = z. Then
dZ (u, v) = dX (u, v).
Z = (Z, eZ, dZ) Î S[A]and the system (Z, s ) is an equalizer of the couple (f, g) in the category S[A].
From the properties (1) and (2) it follows that the following conditions hold:
The next result was established in [9] and it presents in a synthetical form new properties of the category S[A] if A is a complete D-algebra.
3.10 Theorem
If A ÎD* is a biresiduated algebra associated with a complete D-algebra then S[A] is a cartesian closed category.
Theorem 3.10 follows from 3.9(3) and the fact that for all X, YÎS[A], if A is associated with a complete D-algebra then there exists a standard structure of A-valued space on the set Hom(X, Y) and one can define a functor
EX = ( )X : S[A] ®S[A]as follows, for all YÎS[A], g ÎHom(Y, Z)) and f ÎEX(Y)):
· EX(g)(f) = g × f
P[X] = ( ) ´ X : S[A] ®S[A].If A is associated with a complete D-algebra one can prove also that the inverse limits exist in the category S[A].
In order to elaborate a new interesting and comprehensive mathematical theory of fuzzy sets having an unitary logical foundation, different new ways to make the class S[A] into a category can be considered. For this purpose, a good source of inspiration is the elementary toposes theory together with complete W -sets theory, where W is a complete Heyting algebra [5]. A first goal is to identify an adequate class of many-valued mathematical models including sheaves. A completeness theorem with respect to these models also must be proved.
REFERENCES
