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@ Appl. Gen. Topol. 19, no. 1 (2018), 129-144doi:10.4995/agt.2018.7849
c© AGT, UPV, 2018

Characterization of quantale-valued metric

spaces and quantale-valued partial metric

spaces by convergence

Gunther Jäger
a
and T. M. G. Ahsanullah

b

a
School of Mechanical Engineering, University of Applied Sciences Stralsund, 18435 Stralsund,

Germany (gunther.jaeger@hochschule-stralsund.de)
b
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi

Arabia. (tmga1@ksu.edu.sa)

Communicated by V. Gregori

Abstract

We identify two categories of quantale-valued convergence tower spaces

that are isomorphic to the categories of quantale-valued metric spaces

and quantale-valued partial metric spaces, respectively. As an applica-

tion we state a quantale-valued metrization theorem for quantale-valued

convergence tower groups.

2010 MSC: 54A20; 54A40; 54E35; 54E70.

Keywords: L-metric space; L-partial metric space; L-convergence tower

space; L-convergence tower group; metrization.

1. Introduction

There are different generalizations of metric spaces. One of them solves the
problem of assigning a precise value to the distance between two points by
allowing instead the assignment of a probability distribution for each pair of
points, the value of which at u ∈ [0,∞] giving the probabilty that the distance
between the points is less than u. A thorough treatment of these probabilistic
metric spaces can be found in [28]. From a different perspective, metric spaces
are viewed as categories in [19] and later, in [9] it has been shown that not only

Received 05 July 2017 – Accepted 10 October 2017

http://dx.doi.org/10.4995/agt.2018.7849


G. Jäger and T. M. G. Ahsanullah

classical metric spaces but also probabilistic metric spaces are special instances
of this approach. The main idea here is to replace the interval [0,∞] as the
codomain of a metric by a quantale. For this reason, we speak of quantale-
valued metric spaces. Another generalization of metric spaces deals with the
problem, that self-distances are not always zero. Relaxing this requirement
and at the same time enforcing the transivity axiom leads to the theory of
partial metric spaces [17, 21]. This concept was independently, and in much
wider generality, introduced under the name M-valued set by Höhle [11], where
the relationship with the general view point of [19] becomes obvious, as also
M-valued sets take their values in a quantale.

All these generalizations allow the introduction of underlying topological
spaces and, in consequence, of a concept of convergence. In this paper, we look
at convergence for quantale-valued metric spaces from a different perspective.
Rather than describing a concept of convergence underlying a quantale-valued
metric space, we are looking for a concept of convergence that characterizes
such spaces. The key point is here to allow different grades of convergence,
where these grades are interpreted as values in the quantale. In this sense, a
filter in a quantale-valued (partial) metric space converges to a point with a
certain grade. We obtain in this way a family of convergence structures on a
set indexed by the quantale. For the unit interval as ”index set” spaces with
such towers of convergence structures were first studied by Richardson and Kent
under the name probabilistic convergence spaces [26]. In a more general setting,
quantale-valued convergence towers are considered in [15]. In this paper, we
identify a set of axioms, such that the quantale-valued (partial) convergence
tower spaces satisfying these axioms can be identified with quantale-valued
(partial) metric spaces.

The paper is organized as follows. In the second section, we collect the
necessary concepts and notions from lattice theory and fix the notation. In
the third section, we study quantale-valued metric spaces and quantale-valued
convergence tower spaces and their relationship. The fourth section then states
an axiom which ensures the isomorphy of quantale-valued metric spaces and
a category of quantale-valued convergence tower spaces satisfying this axiom.
In a similar fashion, in Section 5, it is shown that quantale-valued partial
convergence tower spaces satisfying certain axioms can be used to character-
ize quantale-valued partial metric spaces. Finally, in Section 6, we apply our
results and state a quantale-valued metrization theorem for quantale-valued
convergence tower groups.

2. Preliminaries

Let L be a complete lattice. We assume that L is non-trivial in the sense
that ⊤ 6= ⊥ for the top element ⊤ and the bottom element ⊥. In any complete
lattice L we can define the well-below relation α ✁ β if for all subsets D ⊆ L
such that β ≤

∨

D there is δ ∈ D such that α ≤ δ. Then α ≤ β whenever α✁β
and α ✁

∨

j∈J βj iff α ✁ βi for some i ∈ J. A complete lattice is completely

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Characterization of L-(partial) metric spaces by convergence

distributive (sometimes called constructively completely distributive) if and only
if we have α =

∨

{β : β ✁ α} for any α ∈ L, [25]. (For a more accessible
proof of the equivalence of this condition with the classical concept of complete
distributivity in the presence of the Axiom of Choice see e.g. Theorem 7.2.3 in
[1].) In a completely distributive lattice L, from α ✁ β =

∨

{γ ∈ L : γ ✁ β}
we infer the existence of γ ∈ L such that α✁γ ✁β, i.e. L satisfies the so-called
interpolation property. For more results on lattices we refer to [10].

The triple L = (L,≤,∗), where (L,≤) is a complete lattice, is called a
quantale [27] if (L,∗) is a semigroup, and ∗ is distributive over arbitrary joins,
i.e.

(

∨

i∈J

αi

)

∗ β =
∨

i∈J

(αi ∗ β) and β ∗

(

∨

i∈J

αi

)

=
∨

i∈J

(β ∗ αi).

A quantale L = (L,≤,∗) is called commutative if (L,∗) is a commutative semi-
group and it is called integral if the top element of L acts as the unit, i.e. if
α ∗ ⊤ = ⊤ ∗ α = α for all α ∈ L. A quantale L = (L,≤,∗) is called an MV-
algebra [13], if for all α,β ∈ L we have (α → β) → β = α∨β. In a quantale we
can define an implication operator by

α → β =
∨

{γ ∈ L : α ∗ γ ≤ β}.

Then δ ≤ α → β if and only if δ ∗ α ≤ β.
We consider in this paper only commutative and integral quantales L =

(L,≤,∗) with completely distributive lattices L.

Example 2.1. A triangular norm or t-norm is a binary operation ∗ on the unit
interval [0,1] which is associative, commutative, non-decreasing in each argu-
ment and which has 1 as the unit. The triple L = ([0,1],≤,∗) can be considered
as a quantale if the t-norm is left-continuous. The three most commonly used
(left-continuous) t-norms are:

• the minimum t-norm: α ∗ β = α ∧ β,
• the product t-norm: α ∗ β = α · β,
• the Lukasiewicz t-norm: α ∗ β = (α + β − 1) ∨ 0.

For the minimum t-norm we obtain α → β =

{

⊤ if α ≤ β
β if α > β

. For the

product t-norm we have α → β = β
α
∧ 1 and for the Lukasiewicz t-norm we

have α → β = (1 − α + β) ∧ 1.

Example 2.2. [19] The interval [0,∞] with the opposite order and addition
as the quantale operation α ∗ β = α + β (extended by α + ∞ = ∞ + a = ∞
for all α,β ∈ [0,∞]) is a quantale L = ([0,∞],≥,+), see e.g. [9]. We have here
α → β = (β − α) ∨ 0.

Example 2.3. A function ϕ : [0,∞] −→ [0,1], which is non-decreasing, left-
continuous on (0,∞) – in the sense that for all x ∈ (0,∞) we have ϕ(x) =
supz<x ϕ(z) – and satisfies ϕ(0) = 0 is called a distance distribution function

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G. Jäger and T. M. G. Ahsanullah

[28]. The set of all distance distribution functions is denoted by ∆+. For
example, for each 0 ≤ a ≤ ∞ the functions

εa(x) =

{

0 if 0 ≤ x ≤ a
1 if a < x ≤ ∞

are in ∆+. The set ∆+ is ordered pointwise, i.e. for ϕ,ψ ∈ ∆+ we define ϕ ≤ ψ
if for all x ≥ 0 we have ϕ(x) ≤ ψ(x). The bottom element of ∆+ is then ε∞ and
the top element is ε0. The set ∆

+ with this order then becomes a complete lat-
tice. We note that

∧

i∈I ϕi is in general not the pointwise infimum. It is not dif-
ficult to show that for all x ∈ [0,∞] we have

∧

j∈J ϕj(x) = supz<x infj∈J ϕj(z)

with the pointwise infimum infj∈J ϕj. It is shown in [9] that ∆
+ is completely

distributive.
A binary operation, ∗ : ∆+×∆+ −→ ∆+, which is commutative, associative,

non-decreasing in each place and that satisfies the boundary condition ϕ∗ε0 = ϕ
for all ϕ ∈ ∆+, is called a triangle function [28]. A triangle function is called
sup-continuous [28], if (

∨

i∈I ϕi) ∗ ψ =
∨

i∈I(ϕi ∗ ψ) for all ϕi,ψ ∈ ∆
+, (i ∈ I),

i.e. if L = (∆+,≤,∗) is a quantale.

For a set X, we denote its power set by P(X) and the set of all filters F,G, ...
on X by F(X). The set F(X) is ordered by set inclusion and maximal elements
of F(X) in this order are called ultrafilters. The set of all ultrafilters on X is
denoted by U(X). In particular, for each x ∈ X, the point filter [x] = {A ⊆ X :
x ∈ A} ∈ F(X) is an ultrafilter. If F ∈ F(X) and f : X −→ Y is a mapping,
then we define f(F) ∈ F(Y ) by f(F) = {G ⊆ Y : f(F) ⊆ G for some F ∈ F}.

We assume some familiarity with category theory and refer to the textbooks
[2] and [23] for more details and notation.

3. L-metric spaces as L-convergence tower spaces

For a quantale L = (L,≤,∗), an L-metric space [19, 9] is a pair (X,d) of a
set X and a mapping d : X × X −→ L such that

(LM1) d(x,x) = ⊤ for all x ∈ X (reflexivity);
(LM2) d(x,y) ∗ d(y,z) ≤ d(x,z) for all x,y,z ∈ X (transitivity).

A mapping between two L-metric spaces, f : (X,d) −→ (X′,d′) is called an L-
metric morphism if d(x1,x2) ≤ d

′(f(x1),f(x2)) for all x1,x2 ∈ X. We denote
the category of L-metric spaces with L-metric morphisms by L-MET.

In case L = {0,1}, an L-metric space is a preordered set. If L = ([0,∞],≥,
+), an L-metric space is a quasimetric space. If L = (∆+,≤,∗), an L-metric
space is a probabilistic quasimetric space, see [9].

Let X be a set. A family of mappings c = (cα : F(X) −→ P(X))α∈L which
satisfies the axioms

(LC1) x ∈ cα([x]) for all x ∈ X,α ∈ L;
(LC2) cα(F) ⊆ cα(G) whenever F ≤ G;
(LC3) cβ(F) ⊆ cα(F) whenever α ≤ β;
(LC4) x ∈ c⊥(F) for all x ∈ X,F ∈ F(X);

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Characterization of L-(partial) metric spaces by convergence

is called an L-convergence tower on X and the pair (X,c) is called an L-
convergence tower space. A mapping f : X −→ X′ between the L-convergence
tower spaces (X,c) and (X′,c′), is called continuous if, for all x ∈ X and all
F ∈ F(X), f(x) ∈ c′α(f(F)) whenever x ∈ cα(F). The category of L-convergence
tower spaces with continuous mappings as morphisms is denoted by L-CTS. We
note that an L-convergence tower space is a stratified {0,1}{0,1}L-convergence
tower space in the definition of [15].

For L = {0,1} an L-convergence tower space is a generalized convergence
space [24], for L = ([0,1],≤,∧) we obtain the probabilistic convergence spaces
in the definition of Richardson and Kent [26] and for L = (∆+,≤,∗) we obtain
the probabilistic convergence spaces in [14]. For L = ([0,∞],≥,+) we obtain
- demanding one additional axiom - the limit tower spaces of Brock and Kent
[8]. It follows from [15] that the category L-CTS is topological, Cartesian closed
and extensional.

An L-convergence tower space (X,c) is called pretopological if the axiom

(LCPT)
⋂

i∈I

cα(Fi) ⊆ cα(
∧

i∈I

Fi) whenever α ∈ L and (Fi)i∈I ∈

F(X)I

is satisfied. It is called left-continuous if for all subsets M ⊆ L we have

(LCLC) x ∈ c∨M(F) whenever x ∈ cα(F) for all α ∈ M.

It is called ∗-transitive if

(LCT) x ∈ cα∗β([z]) whenever x ∈ cα([y]) and y ∈ cβ([z]).

Remark 3.1. A left-continuous L-convergence tower c can be identified with a
limit function λ : F(X) −→ LX, where λ(F)(x) =

∨

{α ∈ L : x ∈ cα(F)}.

We call a left-continuous, ∗-transitive and pretopological L-convergence tower
space an L-premetric convergence tower space and denote the category of these
spaces spaces by L-PreMET-CTS.

Proposition 3.2. Let (X,d) ∈ |L-MET|. Define

x ∈ cdα(F) ⇐⇒
∨

F∈F

∧

y∈F

d(x,y) ≥ α.

Then (X,cd) ∈ |L-CTS|.

Proof. (LC1) We have
∨

F∈[x]

∧

y∈F d(x,y) ≥ d(x,x) = ⊤ ≥ α (with {x} ∈ [x]),

and hence x ∈ cdα([x]) for all α ∈ L and all x ∈ X.
(LC2), (LC3) and (LC4) are obvious. �

Proposition 3.3. Let (X,d) ∈ |L-MET|. Then (X,cd) is ∗-transitive, left-con-
tinuous and pretopological.

Proof. For transitivity, we first note that
∨

F∈[y]

∧

a∈F d(x,a) ≤
∨

F∈[y] d(x,y) =

d(x,y) and, using F = {y} we also have
∨

F∈[y]

∧

a∈F d(x,a) ≥ d(x,y). Hence

x ∈ cdα([y]) ⇐⇒ d(x,y) ≥ α and y ∈ c
d
β([z]) ⇐⇒ d(y,z) ≥ β implies

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G. Jäger and T. M. G. Ahsanullah

d(x,z) ≥ d(x,y) ∗ d(y,z) ≥ α ∗ β, i.e. x ∈ cdα∗β([z]). For left-continuity, let

x ∈ cdα(F) for all α ∈ A ⊆ L. Then for all α ∈ A we have
∨

F∈F

∧

y∈F d(x,y) ≥ α

and therefore also
∨

F∈F

∧

y∈F d(x,y) ≥
∨

A, i.e. x ∈ cd∨
A
(F).

To show the pretopologicalness, let x ∈
⋂

j∈J c
d
α(Fj). Then for all j ∈ J

and all ǫ ✁ α there is Fǫj ∈ Fj such that for all yj ∈ F
ǫ
j , d(x,yj) ≥ ǫ. But

then F =
⋃

j∈J F
ǫ
j ∈

∧

j∈J Fj and for all y ∈ F we have d(x,y) ≥ ǫ. Hence
∨

F∈
∧
j∈J

Fj

∧

y∈F d(x,y) ≥ ǫ, i.e. x ∈ c
d
ǫ(
∧

j∈J Fj). This is true for all ǫ✁α and

by left-continuity and using α =
∨

{ǫ : ǫ ✁ α} we obtain x ∈ cdα(
∧

j∈J Fj). �

Proposition 3.4. Let f : (X,d) −→ (X′,d′) be an L-metric morphism. Then

f : (X,cd) −→ (X′,cd
′

) is continuous.

Proof. Let x ∈ cdα(F). Then

α ≤
∨

F∈F

∧

y∈F

d(x,y) ≤
∨

F∈F

∧

y∈F

d′(f(x),f(y))

=
∨

F∈F

∧

y′∈f(F)

d′(f(x),y′) ≤
∨

G∈f(F)

∧

y′∈G

d′(f(x),y′).

Hence f(x) ∈ cd
′

α (f(F)). �

Hence we have a functor from L-MET into L-PreMET-CTS. We note that
this functor is injective on objects. If d 6= d′ then, without loss of generality,
there are x,y ∈ X such that d(x,y) 6≤ d′(x,y). Then x ∈ cd

d(x,y)
([y]) but

x /∈ cd
′

d(x,y)([y]).

Proposition 3.5. Let (X,c) ∈ |L-PreMET-CTS| and define

dc(x,y) =
∨

x∈cα([y])

α.

Then (X,dc) ∈ |L-MET|.

Proof. (LM1) follows from (LC1) and (LM2) follows from the distributivity of
the quantale operation over arbitrary joins and the transitivity as

dc(x,y) ∗ dc(y,z) =
∨

x∈cα([y]),y∈cβ([z])

α ∗ β ≤
∨

x∈cα∗β([z])

α ∗ β ≤ dc(x,z).

�

Proposition 3.6. Let f : (X,c) −→ (X′,c′) be a morphism in L-PreMET-CTS.

Then f : (X,dc) −→ (X′,dc
′

) is an L-metric morphism.

Proof. We have for x,y ∈ X

dc(x,y) =
∨

x∈cα([y])

α ≤
∨

f(x)∈c′α([f(y)])

α = dc
′

(f(x),f(y)).

�

Proposition 3.7. Let (X,d) ∈ |L-MET|. Then d(c
d) = d.

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Characterization of L-(partial) metric spaces by convergence

Proof. We note again that x ∈ cdα([y]) ⇐⇒ d(x,y) ≥ α. Hence d
(cd)(x,y) =

∨

x∈cdα([y])
α =

∨

d(x,y)≥α α = d(x,y). �

Proposition 3.8. Let (X,c) ∈ |L-PreMET-CTS|. Then c
(dc)
α (F) ⊆ cα(F).

Proof. Let x ∈ c
(dc)
α (F) and let ǫ ✁ α. Using the interpolation property, there

is Fǫ ∈ F such that ǫ ✁ δ ✁
∧

y∈Fǫ
dc(x,y) for some δ ∈ L. Hence for all y ∈ Fǫ

we have ǫ ✁
∨

x∈cβ([y])
β, i.e. for all y ∈ Fǫ there is β ≥ ǫ such that x ∈ cβ([y]).

But then also x ∈ cǫ([y]). Hence we have for all ǫ ✁ α a set Fǫ ∈ F such that
x ∈

⋂

y∈Fǫ
cǫ([y]). Now from Fǫ ∈ F we conclude [Fǫ] =

∧

y∈Fǫ
[y] ≤ F and

from pretopologicalness we conclude that for all ǫ✁α there is Fǫ ∈ F such that
x ∈ cǫ([Fǫ]) ⊆ cǫ(F). This is true for all ǫ ✁ α and from the left-continuity we
again conclude x ∈ c∨{ǫ✁α}(F) = cα(F). �

Theorem 3.9. The category L-MET can be coreflectively embedded into the
category L-PreMET-CTS.

Remark 3.10. In [14] for the case L = (∆+,≤,∗) we embedded the category of
probabilistic metric spaces into the category of probabilistic convergence tower
spaces in a different way. Following Tardiff [29], we define for an L-metric space
(X,d), ǫ > 0 and ϕ ∈ ∆+ the (ϕ,ǫ)-neighbourhood of x ∈ X by

Nϕ,ǫx = {y ∈ X : d(x,y)(u + ǫ) + ǫ ≥ ϕ(u) ∀u ∈ [0,
1

ǫ
)}.

and define the ϕ-neighbourhood filter of x ∈ X, Nϕx as the filter generated by
the sets Nϕ,ǫx , ǫ > 0. If we define

x ∈ c̃dϕ(F) ⇐⇒ F ≥ N
ϕ
x

then we obtain a left-continuous and pretopological L-convergence tower space

(X,c̃d). In order to show that this L-convergence tower space coincides with

the L-convergence tower space (X,cd), we need the following results from [29].
For ϕ ∈ ∆+ and 0 ≤ ǫ ≤ 1 we define ϕǫ ∈ ∆+ by

ϕǫ(u) =







0 if u = 0
(ϕ(u + ǫ) + ǫ) ∧ 1 if 0 < u ≤ 1

ǫ

1 if u > 1
ǫ
.

Clearly then ϕ ≤ ϕǫ and Tardiff [29] shows that y ∈ Nϕ,ǫx if and only if
d(x,y)ǫ ≥ ϕ and ϕ ≥ ψ if and only if for all ǫ > 0 we have ϕǫ ≥ ψ. The last
assertion implies that for ϕ ∈ ∆+ we have ϕ =

∧

ǫ>0 ϕ
ǫ. We will need the

following results.

Lemma A. Let ϕj ∈ ∆
+ for all j ∈ J and let 0 ≤ ǫ ≤ 1. Then (

∨

j∈J ϕj)
ǫ =

∨

j∈J(ϕ
ǫ
j) and (

∧

j∈J ϕj)
ǫ =

∧

j∈J(ϕ
ǫ
j).

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G. Jäger and T. M. G. Ahsanullah

Proof. We only show the second assertion, the first one being similar. For u = 0
or u > 1

ǫ
the assertion is obvious. Let 0 < u ≤ 1

ǫ
. Then we have

(
∧

j∈J

ϕj)
ǫ(u) =



(
∧

j∈J

ϕj)(u + ǫ) + ǫ



 ∧ 1

=

(

( sup
v<u+ǫ

inf
j∈J

ϕj(v)) + ǫ

)

∧ 1

=

(

sup
v<u+ǫ

(inf
j∈J

ϕj(v) + ǫ)

)

∧ 1

= sup
v<u+ǫ

inf
j∈J

((ϕj(v) + ǫ) ∧ 1)

= sup
w+ǫ<u+ǫ

inf
j∈J

((ϕj(w + ǫ) + ǫ) ∧ 1)

= sup
w<u

inf
j∈J

ϕǫj(w)

=
∧

j∈J

ϕǫj(u).

�

Lemma B (cf. [20], Proposition 1.8.29). Let U ∈ U(X) be an ultrafilter and
f : X −→ L be a mapping. Then

∨

U∈U

∧

y∈U f(y) =
∧

U∈U

∨

y∈U f(y).

Proof. It is easy to show that for U,V ∈ U we have
∧

y∈U f(y) ≤
∨

y∈V f(y)

and hence
∨

U∈U

∧

y∈U f(y) ≤
∧

U∈U

∨

y∈U f(y). For the converse inequality,

let α✁
∧

U∈U

∨

y∈U f(y). Then for all V ∈ U there is y ∈ V such that f(y) ≥ α.

With Vα = {y ∈ X : f(y) ≥ α} then V ∩ Vα 6= ∅ for all V ∈ U and hence
Vα ∈ U. Therefore

∨

V ∈U

∧

y∈V f(y) ≥
∧

y∈Vα
f(y) ≥ α. By the complete

distributivity then
∧

U∈U

∨

y∈U f(y) ≤
∨

V ∈U

∧

y∈V f(y). �

Let now x ∈ cdϕ(F) and U ≥ F be an ultrafilter and ψ ✁ ϕ. Then, by Lemma
B,

∧

U∈U

∨

y∈U

d(x,y) =
∨

U∈U

∧

y∈U

d(x,y) ≥ ϕ ✄ ψ,

and hence, for all U ∈ U there is yψ ∈ U such that d(x,yψ)ǫ ≥ d(x,yψ) ≥ ψ.
But this means yψ ∈ Nψ,ǫx for all ǫ > 0. So we conclude that for all U ∈ U we
have Nψ,ǫx ∩ U 6= ∅, and hence, U being an ultrafilter, N

ψ,ǫ
x ∈ U for all ǫ > 0,

i.e. Nψx ≤ U. Therefore x ∈ c̃
d
ψ(U) for all ψ ✁ ϕ and from the left-continuity

then also x ∈ c̃dϕ(U). This is true for all ultrafilters U ≥ F and hence, by

pretopologicalness, x ∈ c̃dϕ(F).
Conversely, let F ≥ Nϕx. Then, for ǫ > 0, we have

∨

F∈F

∧

y∈F

d(x,y)ǫ ≥
∧

y∈N
ϕ,ǫ
x

d(x,y)ǫ ≥ ϕ.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 136



Characterization of L-(partial) metric spaces by convergence

From Lemma A we conclude

ϕ ≤
∧

ǫ>0

∨

F∈F

∧

y∈F

d(x,y)ǫ

=
∧

ǫ>0

(
∨

F∈F

∧

y∈F

d(x,y))ǫ

=
∨

F∈F

∧

y∈F

d(x,y)

and we have x ∈ cdϕ(F).

4. The isomorphy of L-MET and L-MET-CTS

We introduce the following axiom for an L-convergence tower space. We
say that (X,c) ∈ |L-CTS| satisfies the axiom (LM) if for all U ∈ U(X) and all
α ∈ L we have

(LM) x ∈ cα(U) ⇐⇒ ∀U ∈ U,β ✁ α∃y ∈ U s.t. x ∈ cβ([y]).

This axiom was introduced in [7] for probabilistic convergence spaces in the
sense of Richardson and Kent [26].

Theorem 4.1. Let (X,d) ∈ |L-MET|. Then (X,cd) satisfies (LM).

Proof. Let U ∈ U(X) and let α ∈ L. Let first x ∈ cdα(U) and let U ∈ U and
β ✁ α. Then there is Fβ ∈ U such that for all y ∈ Fβ we have d(x,y) ≥ β.
Choose y ∈ U ∩ Fβ. Then

∨

F∈[y]

∧

z∈F

d(x,z) ≥
∧

z∈U∩Fβ

d(x,z) ≥ β,

i.e. x ∈ cβ([y]).
Conversely, let for all U ∈ U, β✁α there is y = yβ ∈ U such that x ∈ c

d
β([y]),

i.e. such that
∨

F∈[y]

∧

z∈F d(x,z) ≥ β. Let further F ∈ U
x =

∧

x∈cd
β
(F) F.

Then, for U ∈ U in particular F ∈ [y], i.e. y ∈ F ∩ U. Hence U ∨ Ux exists and

because U is an ultrafilter, we get U ≥
∧

x∈cd
β
(F) F. As (X,c

d) is pretopological,

we conclude cdβ(U) ⊇
⋂

x∈cd
β
(F) c

d
β(F) and we have x ∈ c

d
β(U). This is true for

any β ✁ α and by left-continuity we obtain x ∈ cdα(U). �

Proposition 4.2. Let (X,c) ∈ |L-PreMET-CTS| satisfy the axiom (LM). Then

c
(dc)
α (F) = cα(F).

Proof. Let U ∈ U(X) be an ultrafilter and let x ∈ cα(U). By the axiom (LM)
we obtain, for β ✁ α that Nxβ = {y ∈ X : x ∈ cβ([y])} satisfies N

x
β ∩ U 6= ∅

for all U ∈ U and hence Nxβ ∈ U. Furthermore, for x ∈ cβ([y]) we have

dc(x,y) ≥ β. Hence
∨

U∈U

∧

y∈U

dc(x,y) ≥
∧

y∈Nx
β

dc(x,y) =
∧

x∈cβ([y])

dc(x,y) ≥ β.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 137



G. Jäger and T. M. G. Ahsanullah

This is true for all β✁α and hence also
∧

y∈Nx
β
dc(x,y) =

∧

x∈cβ([y])
dc(x,y) ≥ α,

which is equivalent to x ∈ c
(dc)
α (U). Hence we have shown cα(U) ⊆ c

(dc)
α (U)

for all U ∈ U(X) and because both (X,c) and (X,c(d
c)) are pretopological, we

have for F ∈ F(X) that cα(F) ⊆ c
(dc)
α (F). The converse implication is always

true and so we have equality. �

If we denote the subcategory of L-PreMET-CTS with objects the L-metric
spaces that satisfy the axiom (LM) by L-MET-CTS, then we obtain the following
main result.

Theorem 4.3. The categories L-MET-CTS and L-MET are isomorphic.

Remark 4.4. For L = (∆+,≤,∗) with a continuous triangle function ∗, we
introduced in [14] for (X,d) ∈ |L-CTS| a different axiom (PM): For all U ∈
U(X), all ϕ ∈ ∆+ and all x ∈ X we have

x ∈ cϕ(U) ⇐⇒ ∀U ∈ U,ǫ > 0 ∃y ∈ U s.t.
∨

x∈cψ([y])

ψ(u+ǫ)+ǫ ≥ ϕ(u)∀u ∈ [0,
1

ǫ
).

With the notation of this paper and of Remark 3.10 then d(c̃
d) = d and if (X,c)

is ∗-transitive, left-continuous and pretopological, then c̃
(dc)
ϕ (F) = cϕ(F) for all

ϕ ∈ ∆+ and all F ∈ F(X). It follows from this, that for an L-convergence tower
space (X,c) that is ∗-transitive, left-continuous and pretopological, the axioms

(PM) and (LM) are equivalent. In fact, if (PM) is true, then c̃
(dc)
ϕ = cϕ and

hence, using Remark 3.10, then also c
(dc)
ϕ = cϕ and as d

c is an L-metric on X

we know that (X,c) = (X,c(d
c)) satisfies (LM). A similar argument shows that

(LM) implies (PM).

5. L-partial metric spaces as L-convergence tower spaces

An L-partial metric space [17, 22] is a pair (X,p) of a set X and a mapping
p : X × X −→ L with

(LPM1) p(x,y) ≤ p(x,x) for all x,y ∈ X;
(LPM2) p(x,y) = p(y,x) for all x,y ∈ X;
(LPM3) p(x,y) ∗ (p(y,y) → p(y,z)) ≤ p(x,z).

Morphisms are defined as in L-MET and the category of L-partial metric spaces
is denoted by L-PMET.

For L = ([0,∞],≥,+), an L-partial metric space is a partial metric space
[22]. These spaces were introduced motivated by problems in computer science,
where the self-distances d(x,x) are not always zero, [21]. Independently and
in much wider generality, L-partial metric spaces were introduced and studied
under the name M-valued sets in [11, 12]. For L = (∆+,≤,∗), L-partial metric
spaces are called probabilistic partial metric spaces in [31] and fuzzy partial
metric spaces in [30].

We note that p(y,z) ≤ p(y,y) → p(y,z) and hence (LPM3) implies the
transitivity axiom (LM2).

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 138



Characterization of L-(partial) metric spaces by convergence

In the sequel, we need to adapt the definition of L-convergence tower spaces.
We relax the axiom (LC1) and replace it by

(wLC1) x ∈ cα([x]) whenever cα([x]) 6= ∅.

An L-partial convergence tower space is a pair (X,c) which satisfies the axioms
(wLC1), (LC2), (LC3) and (LC4). With morphisms as defined before, we
denote the category of L-partial convergence tower spaces by L-PCTS.

We will use the same functors as defined above to embed the category of
L-partial metric spaces into the category of L-partial convergence tower spaces.
Only few adaptations are necessary, so that we simply repeat the results and
only prove the modifications.

Proposition 5.1. Let (X,p) ∈ |L-PMET|. Define

x ∈ cpα(F) ⇐⇒
∨

F∈F

∧

y∈F

p(x,y) ≥ α.

Then (X,cp) ∈ |L-PCTS|.

Proof. We only need to prove (wLC1). As before, we can show that y ∈ cpα([x])
if and only if p(x,y) ≥ α. If y ∈ cpα([x]), then p(x,y) ≥ α and then by (LPM1)
we have p(x,x) ≥ α, i.e. x ∈ cpα([x]). �

We call an L-(partial) convergence tower space (X,c) strongly ∗-transitive if

(LST) x ∈ cα∗(E(y)→β)([z]) whenever x ∈ cα([y]) and y ∈ cβ([z]),

where E(y) =
∨

y∈cγ([y])
γ. It is called symmetric if

(LS) x ∈ cα([y]) whenever y ∈ cα([x]).

Lemma 5.2. If the L-(partial) convergence tower space (X,c) is strongly ∗-
transitive, then it is transitive.

Proof. This follows from α ∗ β ≤ α ∗ (E(y) → β) and the axiom (LC3). �

Proposition 5.3. Let (X,p) ∈ |L-PMET|. Then (X,cp) is strongly ∗-transitive,
left-continuous, symmetric and pretopological.

Proof. We need to check the strong ∗-transitivity (LST) and the symmetry
(LS). For (LST) we first note that E(y) =

∨

y∈c
p

β
([y]) β =

∨

β≤p(y,y) β = p(y,y).

Let x ∈ cpα([y]) and y ∈ c
p
β
([z]). Then α ≤ p(x,y) and β ≤ p(y,z) and

hence α ∗ (E(y) → β) ≤ p(x,y) ∗ (p(y,y) → p(y,z)) ≤ p(x,z) and hence
x ∈ pα∗(E(y)→β)([z]). For (LS), let x ∈ c

p
α([y]). Then p(x,y) = p(y,x) ≥ α and

hence y ∈ cpα([x]). �

Proposition 5.4. Let f : (X,d) −→ (X′,d′) be an L-PMET-morphism. Then

f : (X,cd) −→ (X′,cd
′

) is continuous.

Hence we have a functor from L-PMET into the category of strongly ∗-
transitive, left-continuous, symmetric and pretopological L-partial convergence
tower spaces, L-PrePMET-PCTS. Again, this functor is injective on objects.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 139



G. Jäger and T. M. G. Ahsanullah

In the sequel, we have to restrict the lattice context quite strongly. We say
that the quantale L = (L,≤,∗) satisfies the axiom (DM2) if for all non-empty
index sets J we have
(DM2) α →

∨

j∈J βj =
∨

j∈J(α → βj) for all α,βj ∈ L(j ∈ J).

Examples for quantales that satisfy (DM2) are complete MV-algebras and
also L = ([0,∞],≥,+). In general, L = (∆+,≤,∗) does not satisfy (DM2). We
show this with the following example.

Example 5.5. We consider the triangle function induced by the product t-
norm defined by ϕ ∗ ψ(u) = ϕ(u) · ψ(u) for all u ∈ [0,∞]. We define for each
natural number n ∈ IN the distance distribution function ϕn ∈ ∆

+ by ϕn(u) =
n(u−1) for 1 ≤ u ≤ 1+ 1

n
. Then

∨

n∈IN
ϕn = ε1 and hence ε1 →

∨

n∈IN
ϕn = ε0.

On the other hand it is not difficult to show that ε1 → ϕn = ϕn and hence
∨

n∈IN
(ε1 → ϕn) = ε1.

Proposition 5.6. Let the quantale L = (L,≤,∗) satisfy the axiom (DM2). Let
(X,c) ∈ |L-PrePMET-PCTS| and define

pc(x,y) =
∨

x∈cα([y])

α.

Then (X,pc) ∈ |L-PMET|.

Proof. (LPM1) We have, using (wLC1),

pc(x,y) =
∨

y∈cα([x])

α =

{

⊥ if cα([x]) = ∅
≤
∨

x∈cα([x])
α if cα([x]) 6= ∅

}

≤ pc(x,x).

(LPM2) follows from the symmetry (LS). We show (LPM3). First we note that
E(y) =

∨

y∈cβ([y])
β = pc(y,y). Let now x ∈ cα([y]) and y ∈ cβ([z]). With the

axiom (LST) then x ∈ cα∗(E(y)→β)([z]) and hence α ∗ (E(y) → β) ≤ p
c(x,z).

We conclude with (DM2) and by the distributivity of the quantale operation
over joins

∨

x∈cα([y])

∨

y∈cβ([z])

(α∗(E(y) → β)) = (
∨

x∈cα([y])

α)∗



E(y) →
∨

y∈cβ([z])

β



 ≤ pc(x,z),

which is nothing else than pc(x,y) ∗ (pc(y,y) → pc(y,z)) ≤ pc(x,z). �

Proposition 5.7. Let the quantale L satisfy the axiom (DM2). Let f : (X,c) −→

(X′,c′) be continuous. Then f : (X,pc) −→ (X′,pc
′

) is an L-PMET-morphism.

Proposition 5.8. Let the quantale L satisfy the axiom (DM2). Let (X,p) ∈
|L-PMET|. Then p(c

p) = p.

Proposition 5.9. Let the quantale L satisfy the axiom (DM2). Let (X,c) ∈

|L-PMET-PCTS|. Then c
(pc)
α (F) ⊆ cα(F).

Theorem 5.10. Let the quantale L satisfy the axiom (DM2). Then the category
L-PMET can be coreflectively embedded into the category L-PrePMET-PCTS.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 140



Characterization of L-(partial) metric spaces by convergence

We extend the axiom (LM) to L-partial convergence tower spaces.

(LM) ∀U ∈ U(X),α ∈ L we have
x ∈ cα(U) ⇐⇒ ∀U ∈ U,β ✁ α∃y ∈ U s.t. x ∈ cβ([y]).

The proofs of the following results do not make use of the axiom (LC1) and
hence they carry over to L-partial metric spaces without any alterations.

Theorem 5.11. Let (X,p) ∈ |L-PMET|. Then (X,cp) satisfies (LM).

Proposition 5.12. Let the quantale L satisfy the axiom (DM2) and let (X,c) ∈

|L-PrePMET-PCTS| satisfy the axiom (LM). Then c
(pc)
α (F) = cα(F).

If we denote the subcategory of L-PrePMET-PCTS with objects the L-partial
metric spaces that satisfy the axiom (LM) by L-PMET-PCTS, then we obtain
the following main result.

Theorem 5.13. Let the quantale L satisfy the axiom (DM2). Then the cate-
gories L-PMET-PCTS and L-PMET are isomorphic.

6. L-metrization of L-convergence tower groups

Let (X, ·) be a group with neutral element e. For filters F,G ∈ F(X), the
filter F ⊙ G is generated by the sets F ⊙ G = {xy : x ∈ F,y ∈ G} for F ∈ F
and G ∈ G and the filter F−1 is generated by the sets F−1 = {x−1 : x ∈ F}
for F ∈ F.

Definition 6.1 (see [4]). A triple (X, ·,c), where (X, ·) is a group and (X,c)
is an L-convergence tower space, is called an L-convergence tower group if for
all x,y ∈ X and all F,G ∈ F(X)

(LCTGM) xy ∈ cα∗β(F ⊙ G) whenever x ∈ cα(F) and y ∈ cβ(G);
(LCTGI) x−1 ∈ cα(F

−1) whenever x ∈ cα(F).

A mapping f : X −→ X′, where (X, ·,c) and (X′, ·′,c′) are L-convergence tower
groups, is called an L-CTG-morphism, if f is a homomorphism between the
groups (X, ·),(X′, ·′) and a morphism in L-CTS. The category of L-convergence
tower groups and L-CTG-morphisms is denoted by L-CTG.

For L = {0,1}, we obtain classical convergence groups [18, 6], for L =
([0,1],≤,∗) we obtain the probabilistic convergence groups in the sense of [16]
and for L = (∆+,≤,∗) we obtain the probabilistic convergence groups of [4].
For L = ([0,∞],≥,+) we obtain limit tower groups [3]. An L-convergence tower
group is a stratified {0,1}{0,1}L-convergence tower group in the definition of
[5].

Lemma 6.2. Let (X, ·,c) ∈ |L-CTG| and let α ∈ L, x ∈ X and F ∈ F(X).
Then x ∈ cα(F) if and only if e ∈ cα([x

−1] ⊙ F).

Proof. If x ∈ cα(F) then by (LC1) and (LCTGM) we conclude e = x
−1x ∈

c⊤∗α([x
−1] ⊙ F) = cα([x

−1] ⊙ F). Conversely, if e ∈ cα([x
−1] ⊙ F), then x =

xe ∈ c⊤∗α([x] ⊙ [x
−1] ⊙ F) = cα(F). �

Lemma 6.3. Let (X, ·,c) ∈ |L-CTG|. Then (X,c) is ∗-transitive.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 141



G. Jäger and T. M. G. Ahsanullah

Proof. Let x ∈ cα([y]) and y ∈ cβ([z]). Then e ∈ cα([x
−1] ⊙ [y]) and e ∈

cβ([y
−1] ⊙ [z]). By (LCTGM) then e = ee ∈ cα∗β([x

−1] ⊙ [y] ⊙ [y−1] ⊙ [z]) =
cα∗β([x

−1] ⊙ [z]) and hence x ∈ cα∗β([z]). �

Definition 6.4. A triple (X, ·,d) is called an L-metric group if d is invariant,
i.e. if d(x,y) = d(xz,yz) = d(zx,zy) for all x,y,z ∈ X. A group homomor-
phism f : (X, ·) −→ (X′, ·′) between the L-metric groups (X, ·,d),(X′, ·′,d′) is
called an L-METG-morphism if it is an L-metric morphism between (X,d) and
(X′,d′). The category of L-metric groups is denoted by L-METG.

This definition is motivated by the following result, where we use, for an
L-metric d : X × X −→ L on X, the product L-metric on X × X defined by
d ⊛ d : (X × X) × (X × X) −→ L, d ⊛ d((x,y),(x′,y′)) = d(x,x′) ∗ d(y,y′).

Lemma 6.5. Let (X, ·) be a group and let d : X × X −→ L be an L-metric
which is symmetric, i.e. for which d(x,y) = d(y,x) for all x,y ∈ X holds.
Then the L-metric d is invariant if and only if the mappings m : X ×X −→ X,
m(x,y) = xy and i : X −→ X, i(x) = x−1 are L-metric morphisms.

Proof. Let first d be an invariant metric on X. Then using the transitivity,
we obtain d ⊛ d((x,y),(x′,y′)) = d(x,x′) ∗ d(y,y′) = d(xy,x′y) ∗ d(x′y,x′y′) ≤
d(xy,x′y′) = d(m(x,y),m(x′,y′)), i.e. multiplication is an L-metric morphism.
Furthermore, using the symmetry of d, we obtain d(x,y) = d(y−1xx−1,y−1yx−1)
= d(y−1,x−1) = d(x−1,y−1), i.e. inversion is an L-metric morphism.

For the converse, we note that, multiplication being an L-metric morphism,
we have for all x,x′,y,y′ ∈ X, d(x,y) ∗ d(x′,y′) = d ⊛ d((x,x′),(y,y′)) ≤
d(xy,x′y′). In particular, we have d(x,y) = d(x,y) ∗ d(z,z) ≤ d(xz,yz) and
similarly d(xz,yz) = d(xz,yz) ∗ d(z−1,z−1) ≤ d(xzz−1,yzz−1) = d(x,y). Sim-
ilarly we can show that d(x,y) = d(zx,zy) and hence d is invariant. �

We call an L-convergence tower group (X,c) L-metrizable if there is a sym-

metric and invariant L-metric d on X such that c = cd.

Theorem 6.6. An L-convergence tower group (X, ·,c) is L-metrizable if and
only if it is left-continuous, pretopological, symmetric and satisfies the axiom
(LM).

Proof. We have seen above that if there is an L-metric d such that c = cd, then
(X,c) is left-continuous, pretopological and satisfies the axiom (LM). Symmetry

of (X,cd) follows easily from the symmetry of d. Conversely, let (X,c) be
left-continuous, pretopological, symmetric and satisfy the axiom (LM). Then

dc(x,y) =
∨

x∈cα([y])
α is a symmetric L-metric on X that satisfies cd

c
= c. We

only need to show that dc is invariant. To this end, we note that by (LCTGM)
and (LC1) we have for x,y,z ∈ X that x ∈ cα([y]) if and only if xz ∈ cα([yz]).
Hence dc(xz,yz) =

∨

xz∈cα([yz])
α =

∨

x∈cα([y])
α = dc(x,y). Similarly, we see

that dc(zx,zy) = dc(x,y) and hence dc is invariant. �

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 142



Characterization of L-(partial) metric spaces by convergence

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