Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their
Duality
Mustafa Demirci
Department of Mathematics, Faculty of Sciences, Akdeniz University, 07058, Antalya, Turkey
Keywords:
Fuzzy Set, Fuzzy Logic, Fuzzy Topology, Quantale, Semi-Quantale, Category Theory, Adjoint Situation,
Equivalence, Spatiality, Sobriety.
Abstract:
The present paper introduces M-fuzzy semi-quantales, fuzzifying semi-quantales, and (L, M)-quasi-fuzzy
topological spaces, providing a common framework for (L, M)-fuzzy topological spaces of Kubiak and
ˇ
Sostak,
L-quasi-fuzzy topological spaces of Rodabaugh and L-fuzzy topological spaces of H
¨
ohle and
ˇ
Sostak. In this
paper, we set up a dual adjunction between the category of (L,M)-quasi-fuzzy topological spaces and the
category of M-fuzzy semi-quantales, and then show that this adjunction includes a dual equivalence between
the category of (L, M)-sober (L, M)-quasi-fuzzy topological spaces and the category of (L, M)-spatial M-fuzzy
semi-quantales.
1 INTRODUCTION
Topological duals of ordered algebraic structures have
been a fundamental issue in mathematics (Clark and
Davey, 1998; Ern
´
e, 2004 ; Jhonstone, 1986; Law-
son, 1979) since the seminal paper of M. H. Stone
(Stone, 1936) on the representations of Boolean al-
gebras. Most known fuzzy topological duals of
(fuzzy) ordered algebraic structures in fuzzy math-
ematics (Demirci, 2014; H
¨
ohle, 2001; Solovyov,
2008; Yao, 2012) have been inspired by the famous
Papert-Papert-Isbell adjunction (Isbell, 1972; Jhon-
stone, 1986; Papert and Papert, 1957/1958) between
the category Top of topological spaces and the oppo-
site Frm
op
of the category Frm of frames. (L, M)-
fuzzy topological spaces of Kubiak and Sostak (Ku-
biak and
ˇ
Sostak, 2009), L-quasi-fuzzy topological
spaces of Rodabaugh (Rodabaugh, 2007), L-fuzzy
topological spaces of H
¨
ohle and
ˇ
Sostak (H
¨
ohle and
ˇ
Sostak, 1999) are three important approaches to the
fuzzy topological spaces in which fuzzy topologies
are assumed to be fuzzy sets themselves. In order to
provide a common framework for these approaches,
we propose (L, M)-quasi-fuzzy topological space, and
formulate their category (L, M)-QFTop on the basis
of fixed semi-quantales L and M. Our aim in this pa-
per is to find out suitable (possibly fuzzy) ordered al-
gebraic structures, whose topological counterparts are
(L, M)-quasi-fuzzy topological spaces, and is to set
up an analog of the Papert-Papert-Isbell adjunction
for (L, M)-quasi-fuzzy topological spaces and such
ordered algebraic structures. More clearly, we intro-
duce M-fuzzy semi-quantales as asked ordered alge-
braic structures, and build the category M-FSQuant
of them in the next section. Section 3 is devoted to
(L, M)-quasi-fuzzy topological spaces and their cat-
egories. We reserve the fourth section for the main
contributions of this paper: the adjunction between
(L, M)-QFTop and the dual of M-FSQuant, and the
refinement of this adjunction to a dual equivalence
between the full subcategory of (L, M)-QFTop of
all (L, M)-sober objects and the full subcategory of
M-FSQuant of all (L, M)-spatial objects. This dual
equivalence, providing the representation of M-fuzzy
semi-quantales by means of (L, M)-quasi-fuzzy topo-
logical spaces, can also be thought of as an analog of
the famous Stone duality (Jhonstone, 1986) between
Boolean algebras and compact, Hausdorff and zero-
dimensional topological spaces (alias Stone spaces),
where we replace Boolean algebras and Stone spaces
by (L, M)-spatial M-fuzzy semi-quantales and (L, M)-
sober (L, M)-quasi-fuzzy topological spaces, respec-
tively. In Section 5, we show that the relationships
in Section 4 can also be enlarged to the categories of
strong M-fuzzy semi-quantales and of unital M-fuzzy
semi-quantales.
Demirci, M..
Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their Duality.
In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 2: FCTA, pages 105-111
ISBN: 978-989-758-157-1
Copyright
c
 2015 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
105

2 FUZZY SEMI-QUANTALES
Fuzzy semi-quantale is a fuzzification of the concept
of semi-quantale. A semi-quantale (an s-quantale for
short) (L, ≤, ⊗) is defined to be a complete lattice
(L, ≤) with a binary operation ⊗ : L × L → L, called
a tensor product (Rodabaugh, 2007). As convention,
we denote the join, meet, top and bottom elements
in the complete lattice (L, ≤) by
W
,
V
, >
L
and ⊥
L
,
respectively. S-quantales include various classes of
ordered algebraic structures (e.g., complete residu-
ated lattices, unit interval [0, 1] equipped with uni-
norms (t-norms or t-conorms in particular), quantales,
frames, semi-frames) playing a major role in fuzzy
logics and fuzzy set theory (B
ˇ
elohl
´
avek, 2002; H
´
ajek,
1998; Nov
´
ak, Perfilieva and Mo
ˇ
cko
ˇ
r, 1999). Now we
give only some of their definitions that will be needed
in the following text.
Definition 1. (i) An s-quantale (L, ≤, ⊗) is called a
unital s-quantale, abbreviated as us-quantale if ⊗ has
an identity element e ∈ L called the unit (Rodabaugh,
2007).
(ii) A us-quantale (L, ≤, ⊗) with e = >
L
is called
a strictly two-sided s-quantale, abbreviated as st-s-
quantale (Rodabaugh, 2007).
(iii) An s-quantale (L, ≤, ⊗) is called a complete
groupoid if ⊗ is isotone in both variables (H
¨
ohle,
2001).
(iv) A complete groupoid (L, ≤, ⊗) is called a
complete quasi-monoidal lattice, abbreviated as cqm-
lattice if x ≤ x ⊗ >
L
and x ≤ >
L
⊗ x hold for all x ∈ L
(H
¨
ohle and
ˇ
Sostak, 1999).
(v) An s-quantale (L, ≤, ⊗) is called a quantale
if ⊗ is associative and distributive over arbitrary
W
(Rosenthal, 1990).
All s-quantales constitute a category SQuant
with morphisms (the so-called s-quantale morphisms)
all functions preserving ⊗ and arbitrary
W
(Rod-
abaugh, 2007). An s-quantale morphism is said to
be strong if it preserves the top element (Demirci,
2010). S-quantales together with strong s-quantale
morphisms form a non-full subcategory SSQuant of
SQuant (Demirci, 2010). USQuant is another one
of the non-full subcategories of SQuant with ob-
jects all us-quantales and with morphism (the so-
called us-quantale morphisms) all s-quantale mor-
phisms preserving the unit (Rodabaugh, 2007). Now
we have enough information to introduce fuzzy semi-
quantales:
Definition 2. Let (L, ≤, ⊗) and (M, ≤, ∗) be s-
quantales.
(i) An M-fuzzy semi-quantale on L is a map
µ : L → M satisfying the following conditions: For all
x, y ∈ L and

x
j
| j ∈ J

⊆ L,
(FSQ1) µ(x) ∗ µ (y) ≤ µ (x ⊗ y),
(FSQ2)
V
j∈J
µ(x
j
) ≤ µ
W
j∈J
x
j
!
.
(ii) An M-fuzzy semi-quantale µ is called strong if
µ(>
L
) = >
M
.
(iii) In case (L,≤, ⊗) is a us-quantale with the
unit e
L
, an M-fuzzy semi-quantale µ is called unital
if µ(e
L
) = >
M
.
Throughout this paper, we use the abbreviations
M-fs-quantale, M-fss-quantale and M-fus-quantale
for M-fuzzy semi-quantale, strong M-fuzzy semi-
quantale and unital M-fuzzy semi-quantale, respec-
tively.
Example 3. For a, b ∈ R with a < b and for
any binary operation ? on [a, b], let [a, b]
?
denote
([a, b], ≤, ?), where ≤ is the usual ordering. Further,
let ⊗ and ∗ be binary operations on [0, 1] such that
x ∗ y ≤ x ⊗ y for all x, y ∈ [0, 1].
(i) The identity map id
[0,1]
on [0, 1] is a [0, 1]
∗
-fs-
quantale on [0, 1]
⊗
. In particular, if ⊗ is a t-norm T
(Klement, Mesiar and Pap, 2000), then id
[0,1]
is also
a [0, 1]
∗
-fus-quantale on [0, 1]
T
.
(ii) If ⊗ is the product t-norm T
P
(Klement, Mesiar
and Pap, 2000), then for every nonnegative integer
n, the function [0, 1] → [0, 1], x 7→ x
n
, is a [0, 1]
∗
-fus-
quantale on [0, 1]
T
P
.
(iii) For c, d ∈ R with c < d, let the binary opera-
tion ⊗
1
on [a, b] and the binary operations ∗
1
, ∗
2
on
[c, d] be defined by x ⊗
1
y =
x+y
2
, z ∗
1
w =
z+w
2
and
z ∗
1
w = min
{
z, w
}
.
The linear function [a, b] → [c, d], x 7→ mx + n, is a
[c, d]
∗
i
-fss-quantale on [a, b]
⊗
1
, where i = 1, 2,
m =
d−c
b−a
and n =
b.c−d.a
b−a
.
Definition 4. Let M be an s-quantale.
(i) M-FSQuant is a category that has as objects
all pairs (A, µ) for which A is an s-quantale and µ is
an M-fs-quantale on A, as morphisms from (A
1
, µ
1
)
to (A
2
, µ
2
) all s-quantale morphisms h : A
1
→ A
2
such
that µ
1
(x) ≤ µ
2
(h(x)). Composition and identities are
the same as in the category Set of sets and functions.
(ii) M-FSSQuant is a non-full subcategory of M-
FSQuant in which each object (A, µ) additionally sat-
isfies the property that µ is strong, and each morphism
is additionally strong.
(iii) M-FUSQuant is a non-full subcategory of M-
FSQuant in which each object (A, µ) additionally sat-
isfies the property that A is a us-quantale and µ is uni-
tal, and each morphism is additionally unital.
FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications
106

3 (L, M)-QUASI-FUZZY
TOPOLOGICAL SPACES
Definition 5. Let (L, ≤, ⊗) and (M, ≤, ∗) be s-
quantales, and X a set.
(i) A map τ : L
X
→ M is called an (L, M)-quasi-
fuzzy topology on X iff τ is an M-fs-quantale on L
X
,
i.e. the next conditions are satisfied for all f , g ∈ L
X
and

f
j
| j ∈ J

⊆ L
X
:
(QT1) τ( f ) ∗ τ (g) ≤ τ( f ⊗ g),
(QT2)
V
j∈J
τ( f
j
) ≤ τ
W
j∈J
f
j
!
.
(ii) An (L, M)-quasi-fuzzy topology is strong iff
τ(>
L
X
) = >
M
, where >
L
X
: X → L is the constant
map with value >
L
.
(iii) Let L be a us-quantale with unit e. An (L, M)-
quasi-fuzzy topology is then called an (L, M)-fuzzy
topology iff τ (e
L
X
) = >
M
, where e
L
X
: X → L is the
constant map with value e.
(iv) (X, τ) is called an (L, M)-quasi-fuzzy (resp.
strong (L, M)-quasi-fuzzy, (L, M)-fuzzy) topological
space if τ is an (L, M)-quasi-fuzzy (resp. strong
(L, M)-quasi-fuzzy, (L, M)-fuzzy) topology on X.
Definition 6. Let (L, ≤, ⊗) and (M, ≤, ∗) be s-
quantales.
(i) (L, M)-QFTop denotes a category whose ob-
jects are all (L, M)-quasi-fuzzy topological spaces,
and whose morphisms from (X, τ) to (Y, ν) are all
functions f : X → Y such that ν ≤ τ ◦ f
←
L
, where
f
←
L
: L
Y
→ L
X
is defined by f
←
L
(G) = G ◦ f . Com-
position and identities in (L, M)-QFTop are the same
as in Set.
(ii) (L, M)-SQFTop is a full subcategory of
(L, M)-QFTop consisting of all strong (L, M)-quasi-
fuzzy topological spaces.
(iii) For a us-quantale L, (L, M)-FTop is a
full subcategory of (L, M)-QFTop consisting of all
(L, M)-fuzzy topological spaces.
In case L is an st-s-quantale, (L, M)-SQFTop co-
incides with (L, M)-FTop. As is explained in the
example below, our motivation and the necessity of
(L, M)-(quasi-)fuzzy topological spaces come from
the need for a unification of (L, M)-fuzzy topological
spaces in (Kubiak and
ˇ
Sostak, 2009), L-(quasi-)fuzzy
topological spaces in (Rodabaugh, 2007) and L-fuzzy
topological spaces in (H
¨
ohle and
ˇ
Sostak, 1999).
Example 7. (i) Let (L, ≤, ⊗) and (M, ≤, ∗) be s-
quantales chosen such as ⊗ = ∧, ∗ = ∧ and (M, ≤) is
completely distributive, i.e. the identity
^
i∈I
_
j∈J
i
c
i j
!
=
_
k∈
∏
i∈I
J
i
^
i∈I
c
ik(i)
!
holds for every i ∈ I and

c
i j
| j ∈ J
i

⊆ M. In this
case, (L, M)-FTop (= (L, M)-SQFTop) is the cate-
gory TOP(L, M) of (L,M)-fuzzy topological spaces in
(Kubiak and
ˇ
Sostak, 2009).
(ii) For any s-quantale L, (L, L)-QFTop is the cat-
egory L-QFTop of L-quasi-fuzzy topological spaces
in (Rodabaugh, 2007).
(iii) For any st-s-quantale L, (L, L)-FTop is the
category L-FTop of L-fuzzy topological spaces in
(Rodabaugh, 2007).
(iv) For any cqm-lattice L, (L, L)-SQFTop is the
category L-FTOP of L-fuzzy topological spaces in
(H
¨
ohle and
ˇ
Sostak, 1999).
4 RELATIONS BETWEEN
M-FSQuant AND (L, M)-QFTop
The main objective of this section is to establish a dual
adjunction between (L, M)-QFTop and M-FSQuant
and then to show that this dual adjunction turns into a
dual equivalence between the full subcategory of M-
FSQuant of (L, M)-spatial objects and the full sub-
category of (L, M)-QFTop of (L, M)-sober objects.
In order to realize our aim, we first recall some well-
known category-theoretic tools in the next subsection.
Those already experienced with categories can skip
this subsection. For more details and for all other
category-theoretic notions not explicitly stated in this
paper, we refer the reader to (Ad
´
amek, Herrlich and
Strecker, 1990).
4.1 Adjoint Situations and Equivalences
Our main results in this paper are formulated on the
basis of the notions of adjoint situation, equivalence
and opposite (dual) category. By definition, an adjoint
situation (ρ, φ) : F a G : C → D consists of functors
G : C → D, F : D → C, and natural transformations
id
D
ρ
→ GF (called the unit) and FG
φ
→ id
C
(called the
co-unit) satisfying the adjunction identities
G(φ
A
) ◦ ρ
G(A)
= id
G(A)
and φ
F(B)
◦ F(ρ
B
) = id
F(B)
for all A in C and B in D. If (ρ, φ) : F a G : C → D
is an adjoint situation for some ρ and φ, then F is
said to be a left adjoint to G, F a G in symbols. A
functor G : C → D is called an equivalence if it is full,
faithful and isomorphism-dense. In this case, C and
D are called equivalent categories, denoted by C ∼ D.
Equivalences can also be stated in terms of adjoint
situations: C ∼ D iff there exists an adjoint situation
(ρ, φ) : F a G : C → D with natural isomorphisms ρ
and φ.
Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their Duality
107

The opposite (dual) of a category C is defined
as a category C
op
, whose objects are the same as
C-objects, but morphisms A
u
op
→ B are C-morphisms
B
u
→ A in the opposite direction. Composition of C
op
-
morphisms A
u
op
→ B and B
v
op
→ C is given as A
(u◦v)
op
→ B,
while identities of C and C
op
are the same. We say
that a category C is dually equivalent to another cate-
gory D if C
op
∼ D.
Proposition 8. (Porst and Tholen, 1990) Given an
adjoint situation (ρ, φ) : F a G : C
op
→ D, let Fix (φ)
denote the full subcategory of C of all C-objects A
such that φ
op
A
: A → FGA is a C-isomorphism, and
Fix (ρ) the full subcategory of D of all D-objects B
such that ρ
B
: B → GFB is a D-isomorphism. Then the
restriction of F a G to [Fix (φ)]
op
and Fix (ρ) induces
an equivalence [Fix (φ)]
op
∼ Fix (ρ).
4.2 Adjunction Between M-FSQuant
op
and (L, M)-QFTop
The map S : (L, M)-QFTop → M-FSQuant
op
, de-
fined by
S

(X, τ)
f
−→ (Y, ν)

=

L
X
, τ

(
f
←
L
)
op
−→

L
Y
, ν

,
is clearly a functor. We now construct a functor
T : M-FSQuant
op
→ (L, M)-QFTop with the prop-
erty that S a T by making use of the next lemmas.
Lemma 9. Let (M, ≤, ∗) be a quantale such that
⊥
M
∗ >
M
= >
M
∗ ⊥
M
= ⊥
M
and (M, ≤) is completely distributive.
If h : (A
1
, µ
1
) → (A
2
, µ
2
) is an M-FSQuant-
morphism, then h
→
M
(µ
1
) is an M-fs-quantale on A
2
,
where h
→
M
: M
A
1
→ M
A
2
is defined by
h
→
M
(α)(y) =
_
y=h(x)
α(x) , ∀α ∈ M
A
1
, ∀y ∈ A
2
.
Proof. Let ⊗
i
denote the tensor product on A
i
for
i = 1, 2. In order to see (FSQ1), let us pick arbitrary
y
1
, y
2
∈ A
2
, and say γ = h
→
M
(µ
1
)(y
1
) ∗ h
→
M
(µ
1
)(y
2
). If
h
−1
(y
1
) =
/
0 or h
−1
(y
2
) =
/
0, then
γ = ⊥
M
≤ h
→
M
(µ
1
)(y
1
⊗
2
y
2
).
Suppose h
−1
(y
1
) 6=
/
0 and h
−1
(y
2
) 6=
/
0. By using the
distributivity of ∗ over arbitrary
W
,
γ =


_
y
1
=h(x
1
)
µ
1
(x
1
)


∗


_
y
2
=h(x
2
)
µ
1
(x
2
)


=
_
y
1
=h(x
1
),y
2
=h(x
1
)
µ
1
(x
1
) ∗ µ
1
(x
2
)
≤
_
y
1
=h(x
1
),y
2
=h(x
1
)
µ
1
(x
1
⊗
1
x
2
)
≤
_
y
1
⊗
2
y
2
=h(z)
µ
1
(z) = h
→
M
(µ
1
)(y
1
⊗
2
y
2
).
To prove (FSQ2), let us take an arbitrary

y
j
| j ∈ J

⊆ A
2
. By the complete distributiv-
ity of (M, ≤), we may write
^
j∈J
h
→
M
(µ
1
)(y
j
) =
^
j∈J


_
x∈h
−1
(y
j
)
µ
1
(x)


=
_
k∈
∏
j∈J
h
−1
(y
j
)
^
j∈J
µ
1
(k ( j))
!
≤
_
k∈
∏
j∈J
h
−1
(y
j
)
µ
1
_
j∈J
k ( j)
!!
.
Furthermore, for every k ∈
∏
j∈J
h
−1
(y
j
), since
h(k( j)) = y
j
for every j ∈ J, we easily get
_
j∈J
k ( j) ∈ h
−1
_
j∈J
y
j
!
,
which gives
_
k∈
∏
j∈J
h
−1
(y
j
)
µ
1
_
j∈J
k ( j)
!!
≤
_
z∈h
−1
W
j∈J
y
j
!
µ
1
(z)
= h
→
M
(µ
1
)
_
j∈J
y
j
!
.
Therefore, it follows that
^
j∈J
h
→
M
(µ
1
)(y
j
) ≤ h
→
M
(µ
1
)
_
j∈J
y
j
!
,
and so does the assertion.
Lemma 10. Let L be an s-quantale, and M a quan-
tale with the properties in Lemma 9. For every s-
quantale A, denote by St(A) the set of all s-quantale
morphisms from A to L, and by
h
A
i
the evaluation map
A → L
St(A)
, i.e.
h
A
i
(a)(h) = h (a), ∀a ∈ A, ∀h ∈ St (A) .
FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications
108

Then

St (A), τ
(A,µ)

is an (L, M)-QFTop-object,
where τ
(A,µ)
=
h
A
i
→
M
(µ).
Proof. Consider the constant map θ : L
St(A)
→ M
with value >
M
. Since
h
A
i
: (A, µ) →

L
St(A)
, θ

is
an M-FSQuant-morphism, we directly obtain from
Lemma 9 that τ
(A,µ)
is an M-fs-quantale on L
St(A)
, i.e.

St (A), τ
(A,µ)

is an (L, M)-QFTop-object.
In the remainder of this paper, we always assume
that L is an s-quantale and M is a quantale with the
properties in Lemma 9.
Theorem 11. T : M-FSQuant
op
→ (L, M)-QFTop,
defined by
T

(A
1
, µ
1
)
u
−→ (A
2
, µ
2
)

= T (A
1
, µ
1
)
T (u)
−→ T (A
2
, µ
2
),
is a functor, where T (A, µ) =

St (A), τ
(A,µ)

and
T (u) : St (A
1
) → St (A
2
) is a function, defined by
T (u)(h) = h ◦ u
op
, ∀h ∈ St (A
1
).
Proof. By Lemma 10, T is an object function from
M-FSQuant
op
to (L, M)-QFTop. Since T obviously
preserves composition and identities, we only need
to see that T is a morphism function, i.e. for ev-
ery M-FSQuant
op
-morphism (A
1
, µ
1
)
u
−→ (A
2
, µ
2
),
T (u) : T (A
1
, µ
1
) → T (A
2
, µ
2
) is an (L, M)-QFTop-
morphism, this means that, the inequality
τ
(A
2
,µ
2
)
(G) ≤ τ
(A
1
,µ
1
)
(T (u)
←
L
(G)) (1)
holds for every G ∈ L
St(A
2
)
. In order to prove (1), we
first show that for every a
2
∈ A
2
with G =
h
A
2
i
(a
2
),
T (u)
←
L
(G) =
h
A
1
i
(u
op
(a
2
)). (2)
Given a
2
∈ A
2
, suppose G =
h
A
2
i
(a
2
), i.e. for every
h
2
∈ St (A
2
),
G(h
2
) =
h
A
2
i
(a
2
)(h
2
) = h
2
(a
2
).
Then, for every h
1
∈ St (A
1
), since
T (u)(h
1
) = h
1
◦ u
op
∈ St (A
2
),
and by the definition of T (u)
←
L
, we obtain (2) from
the following observation:
[T (u)
←
L
(G)](h
1
) = G(T (u)(h
1
))
= [T (u)(h
1
)](a
2
)
= h
1
◦ u
op
(a
2
) = h
1
(u
op
(a
2
))
= [
h
A
1
i
(u
op
(a
2
))](h
1
).
On the other hand, since (A
2
, µ
2
)
u
op
−→ (A
1
, µ
1
) is an
M-FSQuant-morphism, we also have
µ
2
(a
2
) ≤ µ
1
(u
op
(a
2
)).
This inequality together with (2) allow us to put down
τ
(A
2
,µ
2
)
(G) =
_
G=
h
A
2
i
(a
2
)
µ
2
(a
2
)
≤
_
T (u)
←
L
(G)=
h
A
1
i
(u
op
(a
2
))
µ
2
(a
2
)
≤
_
T (u)
←
L
(G)=
h
A
1
i
(u
op
(a
2
))
µ
1
(u
op
(a
2
))
≤
_
T (u)
←
L
(G)=
h
A
1
i
(a
1
)
µ
1
(a
1
)
= τ
(A
1
,µ
1
)
(T (u)
←
L
(G)).
Proposition 12. S is left adjoint to T .
Proof. For every (L, M)-QFTop-object (X , τ), define
the function η
(X,τ)
: X → St

L
X

by η
(X,τ)
(x) = π
x
,
where π
x
: L
X
→ L is the x-th projection function for
every x ∈ X.
One can easily see that for every (L, M)-QFTop-
object (X, τ) and every M-FSQuant-object (A, µ),
η
(X,τ)
: (X, τ) → T S (X, τ) and
ε
(A,µ)
=
h
A
i
op
: ST (A, µ) → (A, µ)
are, respectively, an (L, M)-QFTop-morphism and an
M-FSQuant
op
-morphism. Moreover,
η =

η
(X,τ)

: id
(L,M)-QFTop
→ T S and
ε =

ε
(A,µ)

: ST → id
M-FSQuant
op
are natural transformations making
(η, ε) : S a T : M-FSQuant
op
→ (L, M)-QFTop
an adjoint situation, and hence S a T .
4.3 Duality between (L, M)-Spatiality
and (L, M)-Sobriety
We concluded the preceding subsection with the ad-
joint situation
(η, ε) : S a T : M-FSQuant
op
→ (L, M)-QFTop.
With the help of Proposition 8, we restrict, in this sub-
section, this adjoint situation to an equivalence of sub-
categories of M-FSQuant
op
and (L, M)-QFTop in-
volving reciprocal notions of spatiality and sobriety,
which are introduced as follows.
Definition 13. (i) An M-FSQuant-object (A, µ) is
called (L, M)-spatial iff
h
A
i
: (A,µ) → ST (A, µ) is an
M-FSQuant-isomorphism.
(ii) An (L, M)-QFTop-object (X, τ) is called
(L, M)-sober iff η
(X,τ)
: (X, τ) → T S (X, τ) is an
(L, M)-QFTop-isomorphism.
Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their Duality
109

(L, M)-spatiality and (L, M)-sobriety can also be
stated in the following more explicit form:
Proposition 14. (i) An M-FSQuant-object (A, µ) is
(L, M)-spatial iff the evaluation map
h
A
i
: A → L
St(A)
is a bijection, i.e. for every t ∈ L
St(A)
, there exists a
unique a ∈ A such that t =
h
A
i
(a).
(ii) An (L, M)-QFTop-object (X, τ) is (L, M)-
sober iff the projection morphisms π
x
: L
X
→ L are
the only s-quantale morphisms from L
X
to L, i.e. for
every w ∈ St

L
X

, there exists a unique x ∈ X such
that t = π
x
.
Corollary 15. The full subcategory of M-FSQuant
of (L, M)-spatial objects is dually equivalent to the
full subcategory of (L, M)-QFTop of (L, M)-sober ob-
jects.
5 DUALS of M-FSSQuant and
M-FUSQuant
The relationships in Section 4 can also be formulated
for the categories M-FSSQuant and M-FUSQuant
by making some slight changes in the subsections 4.2
and 4.3. To explain how this can be done, we first
fix some notations. For every s-quantale (resp. us-
quantale) A, we denote by St
1
(A) (resp. St
2
(A)) the
set of all strong (resp. unital) s-quantale morphisms
from A to L, where L is assumed to be a us-quantale
in case A is a us-quantale, and by
h
A
i
i
the evaluation
map A → L
St
i
(A)
, i.e.
h
A
i
i
(a)(h) = h (a), ∀a ∈ A, ∀h ∈ St
i
(A)
for i = 1, 2.
It is clear that the restriction of the functor
S : (L, M) -QFTop → M-FSQuant
op
to (L, M)-SQFTop gives a functor
S
1
: (L, M)-SQFTop → M-FSSQuant
op
.
Likewise, for a us-quantale L, S can be restricted to a
functor S
2
: (L, M)-FTop → M-FUSQuant
op
. In the
reverse direction, by following the same steps in The-
orem 11, we may define functors
T
1
: M-FSSQuant
op
→ (L, M)-SQFTop and
T
2
: M-FUSQuant
op
→ (L, M)-FTop
by
T
i

(A, µ)
u
−→ (B, λ)

= T
i
(A, µ)
T
i
(u)
−→ T
i
(B, λ) ,
where i = 1, 2, L is a us-quantale for i = 2,
T
i
(A, µ) =

St
i
(A), τ
(i)
(A,µ)

, τ
(i)
(A,µ)
= (
h
A
i
i
)
→
M
(µ)
and T
i
(u) : St
i
(A) → St
i
(B) is a function given by
T
i
(u)(h) = h ◦ u
op
, ∀h ∈ St
i
(A).
An analogue of Proposition 12 can be expressed
for the functors S
i
and T
i
: S
i
a T
i
(i = 1, 2). The unit η
i
and co-unit ε
i
of the adjunction S
i
a T
i
are the families
η
i
=

η
i
(X,τ)

and ε
i
=

ε
i
(A,µ)

, where i = 1, 2,
η
i
(X,τ)
: (X, τ) → T
i
S
i
(X, τ) and
ε
i
(A,µ)
: S
i
T
i
(A, µ) → (A, µ)
are given by
η
i
(X,τ)
(x) = π
x
and

ε
i
(A,µ)

op
=
h
A
i
i
.
Consequently, (η
i
, ε
i
) : S
i
a T
i
: C
op
i
→ D
i
is an adjoint
situation, where i = 1, 2, L is a us-quantale for i = 2,
C
1
= M-FSSQuant, D
1
= (L, M)-SQFTop,
C
2
= M-FUSQuant, D
2
= (L, M)-FTop.
In a similar fashion to Corollary 15, the adjunction
S
i
a T
i
restricts to an equivalence on the basis of the
following notions of spatiality and sobriety:
Definition 16. (i) An M-FSSQuant-object (A, µ) is
called (L, M)-s-spatial iff
h
A
i
1
: (A, µ) → S
1
T
1
(A, µ)
is an M-FSSQuant-isomorphism.
(ii) An (L, M)-SQFTop-object (X, τ) is called
(L, M)-s-sober iff η
1
(X,τ)
: (X, τ) → T
1
S
1
(X, τ) is an
(L, M)-SQFTop-isomorphism.
Definition 17. Let L be a us-quantale.
(i) An M-FUSQuant-object (A, µ) is called
(L, M)-us-spatial iff
h
A
i
2
: (A, µ) → S
2
T
2
(A, µ) is an
M-FUSQuant-isomorphism.
(ii) An (L, M)-FTop-object (X, τ) is called (L, M)-
us-sober iff η
2
(X,τ)
: (X, τ) → T
2
S
2
(X, τ) is an (L, M)-
FTop-isomorphism.
Corollary 18. (i) The full subcategory of M-
FSSQuant of (L, M)-s-spatial objects is dually equiv-
alent to the full subcategory of (L, M)-SQFTop of
(L, M)-s-sober objects.
(ii) For any us-quantale L, the full subcategory of
M-FUSQuant of (L, M)-us-spatial objects is dually
equivalent to the full subcategory of (L, M)-FTop of
(L, M)-us-sober objects.
FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications
110

6 CONCLUSIONS
In this paper, we have introduced M-fuzzy semi-
quantales and (L, M)-quasi-fuzzy topological spaces.
The former here fuzzify semi-quantales of Rod-
abaugh (Rodabaugh, 2007), while the latter unify
(L, M)-fuzzy topological spaces of Kubiak and Sostak
(Kubiak and
ˇ
Sostak, 2009), L-quasi-fuzzy topological
spaces of Rodabaugh (Rodabaugh, 2007) and L-fuzzy
topological spaces of H
¨
ohle and
ˇ
Sostak (H
¨
ohle and
ˇ
Sostak, 1999). We then have constructed their cat-
egories that enable us to extend the famous Papert-
Papert-Isbell adjunction (Isbell, 1972; Jhonstone,
1986; Papert and Papert, 1957/1958) to the present
setting. One of the central results of this paper is
that there exists a dual adjunction between the cat-
egory of (L, M)-quasi-fuzzy topological spaces and
the category of M-fuzzy semi-quantales. As the other
one, it is shown that this adjunction gives rise to
a dual equivalence between the category of (L, M)-
sober (L, M)-quasi-fuzzy topological spaces and the
category of (L, M)-spatial M-fuzzy semi-quantales.
We finally demonstrate that analogues of these results
are also valid for the categories of strong M-fuzzy
semi-quantales and of unital M-fuzzy semi-quantales.
REFERENCES
Ad
´
amek, J., Herrlich, H. and Strecker, G. E. (1990). Ab-
stract and Concrete Categories. New York: John Wi-
ley & Sons..
B
ˇ
elohl
´
avek, R. (2002). Fuzzy Relational Systems. New
York: Kluwer Academic Publishers.
Clark, D. M. and Davey, B. A. (1998). Natural Dualities for
the Working Algebraist. Cambrige: Cambridge Uni-
versity Press.
Demirci, M. (2010). Pointed Semi-Quantales and Lattice-
Valued Topological Spaces. Fuzzy Sets and Systems,
161, 1224-1241.
Demirci, M. (2014). Fundamental Duality of Abstract Cat-
egories and Its Applications. Fuzzy Sets and Systems,
256, 73-94.
Ern
´
e, M. (2004). General Stone duality. Topology and Its
Applications, 137, 125-158.
H
´
ajek, P. (1998). Metamathematics of Fuzzy Logics. Dor-
drecht: Kluwer Academic Publishers.
H
¨
ohle, U. (2001). Many Valued Topology and Its Applica-
tions. Boston: Kluwer Academic Publishers.
H
¨
ohle, U. and
ˇ
Sostak, A. P. (1999). Axiomatic Founda-
tions of Fixed-Basis Fuzzy Topology. In H
¨
ohle, U. and
Rodabaugh, S. E. (Eds.), Mathematics of Fuzzy Sets:
Logic, Topology and Measure Theory (pp.123-272).
Boston: Kluwer Academic Publishers.
Isbell, J. R. (1972). Atomless Parts of Spaces. Mathematica
Scandinavica, 31, 5-32.
Johnstone, P.T. (1986). Stone Spaces. Cambridge: Cam-
bridge University Press.
Klement, E. P., Mesiar, R. and Pap, E. (2000). Triangular
Norms. Dordrecht: Kluwer Academic Publishers.
Kubiak, T. and
ˇ
Sostak, A. (2009). Foundations of the The-
ory of (L,M)-fuzzy Topological Spaces. In Boden-
hofer, U., DeBaets, B., Klement, E. P. and Saminger-
Platz, S. (Eds.), Abstracts of the 30th Linz Seminar on
Fuzzy Set Theory (pp. 70-73). Linz: Johannes Kepler
Universit
¨
at.
Lawson, J. D. (1979). The Duality of Continuous Posets.
Houston Journal of Mathematics, 5, 357-386. Math. 5
(1979) 357–386.
Nov
´
ak, V., Perfilieva, I. and Mo
ˇ
cko
ˇ
r, J. (1999). Mathe-
matical Principles of Fuzzy Logic. Dordrecht: Kluwer
Academic Publishers.
Papert, D. and Papert, S. (1957/1958). Sur Les Treillis Des
Ouverts Et Les Paratopologies. Seminaire Ehresmann.
Topologie et Geometrie Differentielle, 1, 1-9.
Porst, H. E. and Tholen, W. (1990). Concrete Dualities. In
Herrlich, H. and Porst, H. E. (Eds.), Category Theory
at Work (pp. 111-136). Berlin: Heldermann Verlag.
Rodabaugh, S. E. (2007). Relationship of Algebraic Theo-
ries to Powerset Theories and Fuzzy Topological The-
ories for Lattice-Valued Mathematics. International
Journal of Mathematics and Mathematical Sciences,
71 pages. doi:10.1155/2007/43645.
Rosenthal, K. I. (1990). Quantales and Their Applications.
New York: Longman Scientific and Technical.
Solovyov, S. A. (2008). Sobriety and Spatiality in Varieties
of Algebras. Fuzzy Sets and Systems, 159, 2567-2585.
Stone, M. H. (1936). The Theory of Representations
for Boolean Algebras. Transactions of the American
Mathematical Society, 40, 37-111.
Yao, W. (2012). A Survey of Fuzzifications of Frames, the
Papert-Papert-Isbell Adjunction and Sobriety. Fuzzy
Sets and Systems, 190, 63-81.
Fuzzy Semi-Quantales, (L,M) Quasi-Fuzzy Topological Spaces and Their Duality
111