On some classes of distance distribution function-valued submeasures

Nonlinear Analysis 74 (2011) 1545–1554

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On some classes of distance distribution function-valued submeasures Lenka Halčinová a , Ondrej Hutník a,∗ , Radko Mesiar b,c a

Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01 Košice, Slovakia

b

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia c IRAFM University of Ostrava, 701 03 Ostrava, Czech Republic

article

info

Article history: Received 30 September 2009 Accepted 18 October 2010 MSC: 54E70 60A10 60B05

abstract In this paper we continue the study of a submeasure notion introduced in Hutník and Mesiar (2009) [1] involving a class of operations which provides a generalization of τT -submeasures. We construct pseudo-metrics and metrics generated by such probabilistic submeasures. Two possible generalizations of our submeasure notion are discussed. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Probabilistic metric space Submeasure Triangular norm Menger space Pseudo-metric Generated metric

1. Introduction Let Σ be a ring of subsets of a fixed (non-empty) set Ω and R+ = [0, +∞] be the extended non-negative real half-line. A mapping η : Σ → R+ such that (i) η(∅) = 0; (ii) η(A) ≤ η(B) for A, B ∈ Σ such that A ⊂ B; (iii) η(A ∪ B) ≤ η(A) + η(B) whenever A, B ∈ Σ , is said to be a numerical submeasure on Σ . A pair (Σ , η) will be called a numerical submeasure ring. In paper [1] we have introduced and investigated a submeasure notion related to probabilistic metric spaces (PM-spaces for short) where the triangular norms (t-norms for short) play an essential role. Our considerations of a submeasure notion in paper [1] were closely related to the Menger PM-space (Ω , F , τT ) where τT is the triangle function in the form

τT (F , G)(x) = sup T (F (u), G(v)),

(1)

u+v=x

and T is a left-continuous t-norm. The associated submeasure notion was defined as follows; see Definition 3 in [1]. Definition 1.1. Let T : [0, 1]2 → [0, 1] be a t-norm, and Σ a ring of subsets of Ω ̸= ∅. A mapping γ : Σ → ∆+ (where γ (A) is denoted by γA ) such that (PS1) if A = ∅, then γA (x) = ε0 (x), x > 0;

∗

Corresponding author. E-mail addresses: [email protected] (L. Halčinová), [email protected] (O. Hutník), [email protected] (R. Mesiar).

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.10.026

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(PS2) if A ⊂ B, then γA (x) ≥ γB (x), x > 0; (PS3) γA∪B (x + y) ≥ T (γA (x), γB (y)), x, y > 0, A, B ∈ Σ , is said to be a τT -submeasure and the triple (Σ , γ , T ) is called a τT -submeasure ring. In this paper we provide a generalization of τT -submeasures which involves suitable operations L replacing the standard addition + on R+ such that the underlying function (1) is a triangle function and thus the underlying space is an L-Menger PM-space (for necessary definitions see Section 2). Further we study pseudo-metrics and metrics generated by such τL,T -submeasures. This part was motivated by the results of Chapter 7 in [2] where the problem of quasi-pseudo-metrization of PqpM-spaces is studied. We give some analogous results in connection with τT -submeasures (or its generalizations τL,T -submeasures). Also, some similar results with T -equalities, cf. [3,4] and T -equivalences of type S, cf. [5] are deduced. Finally we discuss possible generalizations of a submeasure notion based on aggregation operators and convolution of distance distribution functions. 2. Preliminary concepts Definition 2.1. Let X be a non-void set. A mapping d : X × X → R+ such that (i) d(x, x) = 0; (ii) d(x, z ) ≤ d(x, y) + d(y, z ) for all x, y, z ∈ X is called a quasi-pseudo-metric on X . If a quasi-pseudo-metric d on X satisfies (iii) d(x, y) = d(y, x) for all x, y ∈ X , then d is called a pseudo-metric on X . A pseudo-metric d on X satisfying (iv) d(x, y) = 0 ⇒ x = y is called a metric on X . Let ∆ be the family of all distribution functions on the extended real line R = [−∞, +∞] (i.e., F : R → [0, 1] is non-decreasing, left-continuous on R, F (−∞) = 0 and F (+∞) = 1) equipped with the usual pointwise order ≤ (i.e., for F , G ∈ ∆, F ≤ G if and only if F (x) ≤ G(x) for all x ∈ R). Denote by εa the unit steps in ∆, i.e., the distribution functions defined for a ∈ [−∞, +∞[ by

εa (x) =

0, 1,



for x ≤ a, for x > a,

while for a = +∞,

ε∞ ( x ) =



0, 1,

for x < +∞, for x = +∞.

A distance distribution function is a distribution function whose support is a subset of R+ , i.e., a distribution function F : R → [0, 1] with F (0) = 0. The class of all distance distribution functions will be denoted by ∆+ , i.e., ∆+ are distribution functions of non-negative random variables. Clearly, εa ∈ ∆+ if and only if a ≥ 0, and ε0 is the maximal element of ∆+ . Distance distribution functions are a proper tool for measuring distances in probabilistic metric spaces [6]. A mapping T : [0, 1]2 → [0, 1] is called a triangular norm (t-norm for short) if it is symmetric, associative, non-decreasing in each argument and has 1 as the identity. The most important t-norms are the minimum M, the product Π , the Łukasiewicz t-norm W and the drastic product D given by M (x, y) := min{x, y};

Π (x, y) := xy; W (x, y) := max{x + y − 1, 0};  min{x, y}, max{x, y} = 1, D(x, y) := 0, otherwise. In the literature these t-norms are often denoted by TM , TP , TL and TD , respectively. For any t-norm T we have M ≥ T ≥ D, in particular, M > Π > W > D; see [7] for comparison of t-norms and [8] for logical, algebraic and probabilistic aspects of t-norms. Just as t-norms lead to the study of associative functions on [0, 1], triangle functions lead to the study of associative functions on ∆+ . Recall that a triangle function is a function τ : ∆+ × ∆+ → ∆+ which is symmetric, associative, nondecreasing in each variable and has ε0 as the identity. Definition 2.2. Let Ω be a non-empty set, F : Ω × Ω → ∆+ a function which assigns to each pair (p, q) ∈ Ω × Ω a distance distribution function Fp,q ∈ ∆+ , and τ : ∆+ × ∆+ → ∆+ a triangle function. The triple (Ω , F , τ ) is called a probabilistic metric space (PM-space for short) if the following properties hold for all p, q, r ∈ Ω : (i) Fp,q = ε0 if and only if p = q;

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(ii) Fp,q = Fq,p ; (iii) Fp,r ≥ τ (Fp,q , Fq,r ). If F satisfies Fp,p = ε0 , p ∈ Ω together with (ii) and (iii), then the triple (Ω , F , τ ) is called a probabilistic pseudo-metric space (PpM-space for short), whereas for F satisfying Fp,p = ε0 , p ∈ Ω with (iii) the triple (Ω , F , τ ) is called a probabilistic quasi-pseudo-metric space (PqpM-space for short). If we deal with possible generalization of τT -submeasures in connection with PM-spaces related to various triangle functions (yielding various geometrical and topological properties of PM-spaces), we have to guarantee that the considered function τ· is indeed an operation on ∆+ and that it is a triangle function on ∆+ . By results in [9] there exists a possibility of such extension of τT -submeasures, see Definition 1.1, to a wider class involving the set of all binary operations L on R+ in (1) such that (L1) (L2) (L3) (L4)

L is commutative and associative; L is jointly strictly increasing, i.e., for all u1 , u2 , v1 , v2 ∈ R+ with u1 < u2 , v1 < v2 holds L(u1 , v1 ) < L(u2 , v2 ); L is continuous on R+ × R+ ; L has 0 as its neutral element.

In fact, L is a semi-group operation on R+ . The usual examples of operations in L are 1

Kα (x, y) := (xα + yα ) α ,

α > 0,

K∞ (x, y) := max{x, y}. Clearly, K1 is the usual sum. Note that min{x, y} ̸∈ L, because (L4) is not satisfied. Moreover, condition (L4) implies the boundedness property, i.e., for each x ∈ R+ the set Ax = {(u, v); L(u, v) = x} is bounded. In general, see [10], L ∈ L if and only if there is a (possibly empty) system (]ak , bk [)k∈K of pairwise disjoint open subintervals of ]0, +∞[, and a system (ℓk )k∈K of increasing bijections ℓk : [ak , bk ] → R+ so that L(x, y) =



1 ℓ− k (ℓk (x) + ℓk (y)), max{x, y},

if (x, y) ∈ ]ak , bk [2 , otherwise.

Note that the set {x ∈ R+ ; L(x, x) = x} of all idempotent elements of L is just the complement of the union of ]ak , bk [. For more details see [7]. By [9, Theorem 7.11] for a left-continuous t-norm T and L ∈ L the function

τL,T (F , G)(x) = sup T (F (u), G(v)) L(u,v)=x

defined for every x ∈ R+ and F , G ∈ ∆+ is a triangle function and the triple (Ω , F , τL,T ) is called an L-Menger PM-space under T . Note that the left-continuity of T is only a sufficient condition, but not necessary for τL,T being a triangle function; see [9, Example 7.1] for T = D and L = K1 . For a continuous t-norm T and L ∈ L the topological semi-groups (R+ , L) and (∆+ , dS , τL,T ) are homeomorphic, cf. [9, Theorem 14.3]. In what follows denote by T the set of all left-continuous t-norms. Then it is naturally to introduce the following generalization of a τT -submeasure notion. Definition 2.3. Let (L, T ) ∈ L × T and Σ be a ring of subsets of Ω ̸= ∅. A mapping γ : Σ → ∆+ satisfying (PS1), (PS2) and (PS4) γA∪B (L(x, y)) ≥ T (γA (x), γB (y)), x, y > 0, A, B ∈ Σ , is said to be a τL,T -submeasure. If L = K1 , then its index is usually omitted and we have the classical τT -submeasure. For L = K∞ we get a τmax,T submeasure related to a non-Archimedean Menger PM-space (Ω , F , τmax,T ). It is worth to note that in this case (PS4) reads as follows

γA∪B (s) ≥ T (γA (s), γB (s)),

s > 0, A, B ∈ Σ .

(2)

Since γ· ∈ ∆ is non-decreasing, then each τL1 ,T -submeasure is a τL2 ,T -submeasure if L1 ≤ L2 . Moreover, if T1 ≥ T2 , then each τL1 ,T1 -submeasure is a τL2 ,T2 -submeasure. +

3. Pseudo-metrics and metrics generated by τL ,T -submeasures 3.1. Menger and L-Menger pseudo-metrics The first statement is related to universal τT -submeasures; see [1, Theorem 2]. In what follows A△B means the symmetric difference of sets A and B.

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Theorem 3.1. Let (Σ , γ , M ) be a τM -submeasure ring, where γA (x) = Φ



x



η(A)

submeasure on Σ . If ρA,B = γA△B , then (Σ , ρ, τM ) is a Menger PpM-space.

, Φ ∈ ∆, A ∈ Σ , x > 0 and η is a numerical

Proof. By [1, Theorem 2] γ is a universal τT -submeasure, therefore ρA,A = ε0 for each A ∈ Σ . Clearly, for A, B ∈ Σ we have ρA,B = ρB,A . Let A, B, C ∈ Σ . Since A△C = (A△B)△(B△C ) ⊂ (A△B) ∪ (B△C )

(3)

and

η(A△C ) ≤ η((A△B) ∪ (B△C )) ≤ η(A△B) + η(B△C ), then for all η(A△C ), η(A△B), η(B△C ) distinct from zero and any x, y > 0 we have x+y

η(A△C )

≥

x+y

η(A△B) + η(B△C )

.

Hence we infer that

 max

x

,



y

η(A△B) η(B△C )

≥



x+y

η(A△B) + η(B△C )

≥ min

x

,

y

η(A△B) η(B△C )



.

From this inequality and the monotonicity of Φ we get

Φ



x+y

η(A△C )



  ≥M Φ

x

η(A△B)



,Φ



y



η(B△C )

,

i.e., ρA,C (x + y) ≥ M (ρA,B (x), ρB,C (y)) for x, y > 0, which means that ρ is a Menger pseudo-metric with respect to M.



In connection with τL,T -submeasures the above result may be generalized as follows. Recall that a set function λ on Σ is translation invariant, if

λ(A, C ) = λ(A△B, B△C ),

A, B, C ∈ Σ .

Theorem 3.2. Let (L, T ) ∈ L × T . If γ is a τL,T -submeasure on Σ , then (Σ , ρ, τL,T ) is an L-Menger PpM-space (under T ). Moreover, ρ is translation invariant. Proof. Clearly, it is enough to prove the Menger triangle inequality. If A, B, C ∈ Σ , then from (3) we have

ρA,C (L(x, y)) = γA△C (L(x, y)) ≥ γ(A△B)∪(B△C ) (L(x, y)) ≥ T (γA△B (x), γB△C (y)) = T (ρA,B (x), ρB,C (y)),

(4)

i.e., ρ is an L-Menger pseudo-metric on Σ . Translation invariance follows from (3) as

ρA,C = γA△C = γ(A△B)△(B△C ) = ρA△B,B△C , which completes the proof.



For a fixed operation L ∈ L and T1 ∈ T denote by ΓL,T1 (Σ , γ ) the set of all L-Menger pseudo-metrics ρ generated by a τL,T1 -submeasure γ on Σ , i.e., ρA,B = γA△B , A, B ∈ Σ . Define a relation ≺ on ΓL,T1 (Σ , γ ) as follows

ρ ≺ ϱ ⇔ ρA,B ≥ ϱA,B for each A, B ∈ Σ . Clearly, ≺ is a partial order on ΓL,T1 (Σ , γ ) and νA,B = ε0 for each A, B ∈ Σ is an element such that ν ≺ ρ for every ρ ∈ ΓL,T1 (Σ , γ ). Let T2 be a t-norm such that T2 dominates T1 (in symbols T2 ≫ T1 ), i.e., T2 (T1 (x, y), T1 (u, v)) ≥ T1 (T2 (x, u), T2 (y, v)) for all x, y, u, v ∈ [0, 1]; see [7]. Define a binary operation ⊕T2 on ΓL,T1 (Σ , γ ) such that for all ρ, ϱ ∈ ΓL,T1 (Σ , γ )

(ρ ⊕T2 ϱ)A,B (z ) = T2 (ρA,B (z ), ϱA,B (z )),

A, B ∈ Σ , z > 0.

We show that ρ ⊕T2 ϱ ∈ ΓL,T1 (Σ , γ ). Indeed, for A ∈ Σ and z > 0 we have

(ρ ⊕T2 ϱ)A,A (z ) = T2 (ρA,A (z ), ϱA,A (z )) = T2 (ε0 (z ), ε0 (z )) = ε0 (z ),

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and the symmetry (ρ ⊕T2 ϱ)A,B = (ρ ⊕T2 ϱ)B,A is obvious for each A, B ∈ Σ . Also, by (4) and domination of T2 over T1 we have

(ρ ⊕T2 ϱ)A,C (L(x, y)) = ≥ ≥ =

T2 (ρA,C (L(x, y)), ϱA,C (L(x, y))) T2 (T1 (ρA,B (x), ρB,C (y)), T1 (ϱA,B (x), ϱB,C (y))) T1 (T2 (ρA,B (x), ϱA,B (x)), T2 (ρB,C (y), ϱB,C (y))) T1 ((ρ ⊕T2 ϱ)A,B (x), (ρ ⊕T2 ϱ)B,C (y)),

where A, B, C ∈ Σ and x, y > 0. Thus ρ ⊕T2 ϱ is an L-Menger pseudo-metric on Σ . Note that for all ρ ∈ ΓL,T1 (Σ , γ ) and T2 ≫ T1

(ν ⊕T2 ρ)A,B (z ) = T2 (νA,B (z ), ρA,B (z )) = T2 (ε0 (z ), ρA,B (z )) = ρA,B (z ) holds for each A, B ∈ Σ and z > 0. The operation ⊕T2 is clearly commutative and associative. Thus we have the following result. Proposition 3.3. Let T2 ≫ T1 . The triple (ΓL,T1 (Σ , γ ), ⊕T2 , ≺) is a partially ordered Abelian semi-group with the neutral element ν . The following theorem provides a relationship between the relation ≺ and the operation ⊕T2 . Theorem 3.4. Let (ΓL,T1 (Σ , γ ), ⊕T2 , ≺) be as in Proposition 3.3. Then for all ρ, ϱ, σ ∈ ΓL,T1 (Σ , γ ) holds ρ ⊕T2 σ ≺ ϱ ⊕T2 σ whenever ρ ≺ ϱ. Proof. The relation ρ ≺ ϱ means that ρA,B ≥ ϱA,B for each A, B ∈ Σ . From monotonicity of T2 we have T2 (ρA,B (z ), σA,B (z )) ≥ T2 (ϱA,B (z ), σA,B (z )) for z > 0. The proof is complete.  Since M ≫ T for each t-norm T and M is the only t-norm whose sets of idempotent elements is [0, 1], cf. [7], we have the following result. Recall that a semi-lattice is an idempotent Abelian semi-group. A semi-lattice is bounded if it includes the neutral element. Theorem 3.5. The ordered pair (ΓL,T1 (Σ , γ ), ⊕M ) is a bounded semi-lattice with the properties (i) ρ ≺ ϱ if and only if ρ ⊕M ϱ = ϱ; (ii) (σ ⊕T2 ρ) ⊕M (σ ⊕T2 ϱ) ≺ σ ⊕T2 (ρ ⊕M ϱ) for each ρ, ϱ, σ ∈ ΓL,T1 (Σ , γ ) and T2 ≫ T1 . Proof. Since ν ∈ ΓL,T1 (Σ , γ ) is the neutral element, the first part of the theorem is obvious. If ρ ≺ ϱ, then ρA,B ≥ ϱA,B for each A, B ∈ Σ , thus ρ ⊕M ϱ = ϱ which shows property (i). For the proof of the second one it is enough to observe that ρ ≺ ρ ⊕M ϱ and ϱ ≺ ρ ⊕M ϱ. Then by Theorem 3.4 we have σ ⊕T2 ρ ≺ σ ⊕T2 (ρ ⊕M ϱ) and σ ⊕T2 ϱ ≺ σ ⊕T2 (ρ ⊕M ϱ). Since (ΓL,T1 (Σ , γ ), ⊕M ) is a semi-lattice, then property (ii) follows.  3.2. Generated pseudo-metrics and metrics In what follows we construct some more pseudo-metrics and metrics on Σ generated by τL,T -submeasures. Easily, [1, Proposition 1] implies the following result. Proposition 3.6. Let L ∈ L be an operation on R+ such that L ≤ K1 . If γ is a τL,T -submeasure on Σ , then the function

χ (A, B) = sup{x > 0; γA△B (x) < 1},

A, B ∈ Σ ,

(5)

is a pseudo-metric on Σ . Observe that if L is Archimedean, i.e., if L(x, x) > x for all x ∈]0, +∞[, then L(x, y) = ℓ−1 (ℓ(x) + ℓ(y)) for some automorphism of R+ (ℓ is unique up to a positive multiplicative constant). Then L ≤ K1 if and only if ℓ is superadditive, i.e., ℓ(x + y) ≥ ℓ(x) + ℓ(y). Note that the convexity of ℓ is enough to ensure its superadditivity. As an example, note that Kα ≤ K1 for any α ∈]1, +∞[ due to the convexity of the corresponding additive generator ℓα (x) = xα . Recall that an additive generator t : [0, 1] → R of a t-norm T is a strictly decreasing function, right-continuous in 0 satisfying t (1) = 0 such that for all (x, y) ∈ [0, 1]2 we have t (x) + t (y) ∈ Ran(t ) ∪ [t (0), +∞], T (x, y) = t (−1) (t (x) + t (y)),

where t (−1) : R → [0, 1] is the pseudo-inverse function to t, i.e., t (−1) (z ) = sup{x ∈ [0, 1]; t (x) > z }. Remark 3.7. We are interested in left-continuous (Archimedean) t-norms and in this case the left-continuity implies continuity; cf. [11]. However, then t is also a continuous additive generator and the formula for its pseudo-inverse can

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be written in an equivalent form t (−1) (z ) =

 1,

t −1 (z ), 0,

z < 0, z ∈ [0, t (0)], z > t (0).

The next result follows from [1, Theorem 4], where we have associated a numerical submeasure to a τT -submeasure for a t-norm having an additive generator. Again, we state it in a slightly more general form. Proposition 3.8. Let L be as in Proposition 3.6. If γ is a τL,T1 -submeasure on Σ and t is an additive generator of a t-norm T such that T ≤ T1 , then

µt (A, B) = sup{x > 0; t (γA△B (x)) ≥ x},

A, B ∈ Σ ,

(6)

is a pseudo-metric on Σ . Immediately we have Theorem 3.9. Let L be as in Proposition 3.6 and γ be a τL,T -submeasure on Σ . If the condition

γA (x) = ε0 (x) ⇒ A = ∅,

x > 0,

(7)

is fulfilled, then χ given by (5) is a metric on Σ . Moreover, if T is a continuous Archimedean t-norm with an additive generator t and γ satisfies (7), then µt given by (6) is also a metric on Σ . Proof. By Propositions 3.6 and 3.8 the functions χ and µt are pseudo-metrics on Σ . Therefore it is enough to prove condition (iv) of Definition 2.1. Let A, B ∈ Σ such that χ (A, B) = 0. Then γA△B (x) = 1 for all x > 0, i.e., γA△B = ε0 and by (7) we have A = B. For µ let A, B ∈ Σ be such that µt (A, B) = 0. Then t (γA△B (x)) < x for all x > 0, i.e., γA△B (x) ≥ t (−1) (x) for all x > 0. Therefore γA△B (0+ ) = 1, hence γA△B = ε0 . Analogously as above we get A = B. The proof is complete.  Remark 3.10. Note that the additional condition (7) is also sufficient to prove that the function ρ in Theorems 3.1 and 3.2 is a Menger and L-Menger metric on Σ , respectively. Thus the corresponding triples (Σ , ρ, τM ) and (Σ , ρ, τL,T ) are the Menger and L-Menger PM-spaces, respectively. Also, in this spirit we could study the families of L-Menger metrics on Σ as above. Remark 3.11. If we replace the constraint L ≤ K1 for the operation L from Propositions 3.6, 3.8 and Theorem 3.9 by the requirement of Archimedean property for L, then, as already mentioned, L admits a representation L(x, y) = ℓ−1 (ℓ(x)+ℓ(y)). For example, ℓα (x) = xα is the additive generator of Kα , α > 0, however K∞ has no additive generator. Therefore, if γ is a τL,T -submeasure on Σ , and T is a continuous Archimedean t-norm with an additive generator t, then the function

µℓ,t (A, B) = ℓ(µt (A, B)),

A, B ∈ Σ

is a (generalized) pseudo-metric on Σ . Moreover, if γ satisfies condition (7), then µℓ,t is a metric on Σ . Remark 3.12. Proposition 3.8 and Theorem 3.9 holds also for the distance functions of Fréchet type of the form

µF (A, B) = inf{x > 0; x + t (γA△B (x))}. Theorem 3.13. Let γ be a τmax,T -submeasure on Σ such that T ≥ W . Then

µ(A, B) =

1

∫

(1 − γA△B (x)) dx,

A, B ∈ Σ ,

(8)

0

is a pseudo-metric on Σ . Proof. It is enough to prove the triangle inequality. From (3) and (2) we have 1 − γA△C (s) ≤ 1 − γ(A△B)∪(B△C ) (s) ≤ 1 − T (γA△B (s), γB△C (s))

≤ 1 − W (γA△B (s), γB△C (s)) = 1 − max{γA△B (s) + γB△C (s) − 1, 0} ≤ 1 − γA△B (s) + 1 − γB△C (s) for each s > 0 and A, B, C ∈ Σ . Integrating over [0, 1] with respect to s we get the desired inequality µ(A, C ) ≤ µ(A, B) + µ(B, C ).  Replacing max with L as in Proposition 3.6 we obtain the first part of the following statement. Since t (x) = 1 − x is an additive generator of W , replacing 1 − γA△B with t (γA△B ) we get its second part.

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Proposition 3.14. Let L be as in Proposition 3.6. (i) If γ is a τL,T -submeasure on Σ such that T ≥ W , then function (8) is a pseudo-metric on Σ . (ii) If γ is a τL,T1 -submeasure on Σ such that t is an additive generator of a t-norm T and T ≤ T1 , then

νt (A, B) =

1

∫

t (γA△B (x)) dx,

A, B ∈ Σ ,

0

is a pseudo-metric on Σ . Remark 3.15. It is easy to verify that if γ satisfies (7), then functions in Theorem 3.13 and Proposition 3.14 are metrics on Σ . In what follows we consider the case L = K1 although the results could be stated in a more general form for L ∈ L. Let (Σ , γ , T ) be a τT -submeasure ring and for each a ∈ [0, 1[ define the function pa : Σ × Σ → R+ by pa (A, B) = inf{x > 0; γA△B (x) > a}.

(9)

Since γ is non-decreasing and left-continuous, then for E , F ∈ Σ and a ∈ [0, 1[ pa (E , F ) < z if and only if γE △F (z ) > a. Clearly, the family of all functions pa satisfies pa (A, A) = 0 for each A ∈ Σ and a ∈ [0, 1[. Also, pa (A, B) = pa (B, A) for A, B ∈ Σ , i.e., pa are semi-pseudo-metrics on Σ . Under the additional assumption on γ we are able to prove the triangle inequality for pa . Theorem 3.16. Let (Σ , γ , T ) be a τT -submeasure ring. If for all a ∈ [0, 1[ the condition

γA△B (x) > a

γB△C (y) > a ⇒ γA△C (x + y) > a,

A, B, C ∈ Σ , x, y > 0,

(10)

is fulfilled, then the function pa is a pseudo-metric on Σ for each a ∈ [0, 1[. Moreover, pa is a metric on Σ for each a ∈]0, 1[ if and only if γA△B (x) > a for all x > 0 whenever A, B ∈ Σ , A ̸= B. Proof. For an arbitrary s > 0 put x = pa (A, B) + γA△C (x + y) > a, and so

s 2

and y = pa (B, C ) + 2s . Then γA△B (x) > a and γB△C (y) > a, which yields

pa (A, C ) < x + y = pa (A, B) + pa (B, C ) + s. Since s is arbitrary, we get the desired inequality pa (A, C ) ≤ pa (A, B) + pa (B, C ). The last statement follows from the fact that pa (A, B) = 0 if and only if γA△B (x) > a for each x > 0.



Corollary 3.17. Let (Σ , γ , T ) be a τT -submeasure ring such that γ satisfies (10). If for each A, B ∈ Σ there exists xAB < +∞ such that γA△B (xAB ) = 1, then the function pa in (9) is a pseudo-metric on Σ for each a ∈ [0, 1], where p1 : Σ × Σ → R+ is given by p1 (A, B) = inf{x > 0; γA△B (x) = 1}, Proof. Let s > 0 and put x = p1 (A, B) + γA△C (x + y) = 1. Therefore we have

A, B ∈ Σ . s 2

, y = p1 (B, C ) + 2s . Then γA△B (x) = 1 and γB△C (y) = 1, which implies that

p1 (A, C ) < x + y = p1 (A, B) + p1 (B, C ) + s. Since s is arbitrary, we get the desired triangle inequality.



Remark 3.18. The family of submeasures satisfying condition (10) is non-empty, because each universal τT -submeasure γ , see [1], satisfies (10). Indeed, if γA△B (x) > a and γB△C (y) > a, then by (3) we get

γA△C (x + y) ≥ γ(A△B)∪(B△C ) (x + y) ≥ M (γA△B (x), γB△C (y)) > M (a, a) = a for A, B, C ∈ Σ and x, y > 0. Recall that a diagonal of a mapping T : [0, 1]2 → [0, 1] is a mapping δT : [0, 1] → [0, 1] given by δT (x) = T (x, x). Denote by F (δT ) the set of all fixed points of δT , i.e., F (δT ) = {x ∈ [0, 1]; δT (x) = x}. Theorem 3.19. Let (Σ , γ , T ) be a τT -submeasure ring and the diagonal δT be strictly increasing and continuous on some interval [a, b[⊂ [0, 1]. If a ∈ F (δT ), then the function pa in (9) is a pseudo-metric on Σ . For a > 0 the function pa is a metric on Σ if and only if γA△B (x) > 0 for all x > 0 whenever A, B ∈ Σ , A ̸= B. Proof. It suffices to show the validity of condition (10) for each a ∈ [0, 1[. Let γA△B (x) > a and γB△C (y) > a, A, B, C ∈ Σ , x, y > 0. Since γ· is non-decreasing and left-continuous, there exists s > 0 such that a + s < b, γA△B (x) > a + s and

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γB△C (y) > a + s. From (3) and the properties of diagonal δT we have γA△C (x + y) ≥ γ(A△B)∪(B△C ) (x + y) ≥ T (γA△B (x), γB△C (y)) ≥ δT (a + s) > δT (a) = a. Now the assertion follows from Theorem 3.16.



Theorem 3.20. Let (Σ , γ , T ) be a τT -submeasure ring with T ≥ Π . If β : R+ → [0, 1] is defined by the formula

β(x) =



e−bx , 0,

x ∈ R+ , x = +∞,

for b ∈]0, +∞[, then the functions qb : Σ × Σ → R+ given by qb (A, B) = inf{x > 0; γA△B (x) > β(x)},

A, B ∈ Σ ,

(11)

are pseudo-metrics on Σ . Proof. Clearly, for each b ∈]0, +∞[ the functions β are strictly decreasing. Let s > 0 and put x = qb (A, B) + qb (B, C ) + 2s , A, B, C ∈ Σ . This means that the following inequalities hold

s 2

,y =

γA△B (x) ≥ β(qb (A, B)) > β(x), γB△C (y) ≥ β(qb (B, C )) > β(y). Then we have

γA△C (x + y) ≥ γ(A△B)∪(B△C ) (x + y) ≥ T (γA△B (x), γB△C (y)) ≥ T (β(qb (A, B)), β(qb (B, C ))) ≥ Π (β(qb (A, B)), β(qb (B, C ))) > Π (β(x), β(y)) = e−bx e−by = e−b(x+y) = β(x + y), which means that qb (A, C ) < x + y = qb (A, B) + qb (B, C ) + s for any s > 0, therefore the triangle inequality holds.



Theorem 3.21. Let (Σ , γ , T ) be a τT -submeasure ring with T ≥ W . If β : R+ → [0, 1] is defined by the formula  x 1 − , x ∈ [0, b], b β(x) = 0, x > b, for b ∈]0, +∞[, then the functions qb in (11) are pseudo-metrics on Σ . Proof. Let s > 0 and put x = qb (A, B) + 2s , y = qb (B, C ) + 2s , A, B, C ∈ Σ . Then

γA△C (x + y) ≥ T (γA△B (x), γB△C (y)) ≥ T (β(qb (A, B)), β(qb (B, C ))) ≥ W (β(qb (A, B)), β(qb (B, C ))) > W (β(x), β(y))   x y x+y = max 1 − + 1 − − 1, 0 = 1 − = β(x + y). b

b

b

Thus qb (A, C ) < x + y = qb (A, B) + qb (B, C ) + s for any s > 0, which implies the desired triangle inequality.



Since each continuous Archimedean t-norm is isomorphic either to Π or to W , the above two theorems imply the following result. Proposition 3.22. Let (Σ , γ , T1 ) be a τT1 -submeasure ring with T1 ≥ T , where T is a continuous Archimedean t-norm with an additive generator t. If β(x) = t (−1) (bx) for x ∈ R+ and b ∈]0, +∞[, then the functions qb in (11) are pseudo-metrics on Σ . 4. Two generalizations 4.1. τL,Q -submeasures Since t-norms are rather special operations on the unit interval [0, 1], we propose to extend our considerations to some particular classes of aggregation operators. An easy approach to generalize τL,T -submeasures is to replace a left-continuous t-norm T with a mapping f : [0, 1]2 → [0, 1] and consider the function

τL,f (F , G)(x) = sup f (F (u), G(v)) L(u,v)=x

in its most general form. Here we consider f = Q , where Q : [0, 1]2 → [0, 1] is a left-continuous binary aggregation operator (i.e., a non-decreasing function in both components with the boundary conditions Q (0, 0) = 0, Q (1, 1) = 1). It is known, see [9, Theorem 6.1], that the left-continuity of Q ensures that τL,Q is a binary operation on ∆+ . However, τL,Q need not be associative in general, but it has good properties on ∆+ . Thus, we may introduce the following notion of submeasure.

L. Halčinová et al. / Nonlinear Analysis 74 (2011) 1545–1554

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Definition 4.1. Let Σ be a ring of subsets of Ω . A mapping γ : Σ → ∆+ satisfying (PS1), (PS2) and (PS5) γA∪B (L(x, y)) ≥ Q (γA (x), γB (y)), x, y > 0, A, B ∈ Σ , is said to be a τL,Q -submeasure. Again, if L = K1 , then its index is usually omitted, and we simply write τQ . Clearly, for Q = T , a t-norm, and L = K1 the τL,Q -submeasure reduces to τT -submeasure from [1]. For L = Kα and Q = T we have

τKα ,T (F , G)(x) = τT (F , G)(xα ). Similarly, by virtue of Remark 3.11 if L is generated by a strictly increasing bijection ℓ : R+ → R+ , we denote L = Kℓ , then

τKℓ ,T (F , G)(x) = τT (F , G)(ℓ(x)). One of the most natural and useful families of aggregation operators is the class of quasi-arithmetic means; see [12]: if f : [0 , 1] → R is a continuous strictly monotone function with {f (0), f (1)} ̸= {−∞, +∞}, then the aggregation operator Mf : n∈N [0, 1]n → [0, 1] given by

 M f ( x1 , . . . , xn ) = f

−1

n 1−

n i =1

 f ( xi )

is called a quasi-arithmetic mean. Of course, we are mainly interested in binary quasi-arithmetic means. In this family the most popular are the arithmetic mean A, the geometric mean G, the harmonic mean H and for p ∈ R \ {0} the p-mean Mp given by, respectively, A(x, y) :=

x+y

G(x, y) :=

√

H(x, y) :=

2 xy;

2xy x+y

Mp (x, y) :=



;

;

xp + y p 2

 1p

.

It is well known that Mf is symmetric, strictly monotone and continuous unless Ran(f ) ̸= R. Moreover, if f (1) is finite (i.e., if 1 is not an annihilator of Mf ), then there is a one-to-one correspondence between the set of continuous Archimedean t-norms and the set of quasi-arithmetic means; see [7]. Indeed, for a fixed L choose Q as a quasi-arithmetic mean generated by an additive generator t (in this spirit, A corresponds to W —a generator of which is given by t (x) = 1 − x, G to Π — p generated by t (x) = − ln x, the p-mean Mp is the counterpart of the Schweizer–Sklar family TpSS —generated by t (x) = 1−px x implying that H is equivalent to the Hamacher product T0H = T−SS1 —generated by t (x) = 1− ) and then x

τL,Q = h(τL,T ) where h(x) = t (−1)



t (x)



2

.

In this light [1, Theorem 3] yields the following result. Proposition 4.2. Let (Σ , η) be a numerical submeasure ring and f : [0, 1] → R be a strictly monotone continuous function such that 1 is not an annihilator of Mf . (i) If f is decreasing, then γ : Σ → ∆+ given by

γA (x) = sup{x ∈ [0, 1]; f (x) > f (1) + η(A) − x} is a τMf -submeasure. (ii) If f is increasing, then

γA (x) = sup{x ∈ [0, 1]; f (x) < f (1) + η(A) − x} is a τMf -submeasure. 4.2. C -based convolution and ∗-submeasures Also, copulas and/or quasi-copulas, cf. [13], may be considered as operators Q . In connection with different triangle functions (and also copulas, more precisely product copula) the usual convolution of distance distribution functions could be used to obtain a submeasure notion corresponding to the Wald space. Indeed, the convolution ∗ on ∆+ is defined as

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follows

(F ∗ G)(x) =

 0,   ∫ x   

x = 0, F (x − t ) dG(t ),

x ∈]0, +∞[,

0

1,

x = +∞,

for each F , G ∈ ∆+ . Then the ∗-submeasure may be defined as follows. Definition 4.3. Let Σ be a ring of subsets of Ω . A mapping γ : Σ → ∆+ satisfying (PS1), (PS2) and (PS6) γA∪B (x) ≥ (γA ∗ γB )(x), x > 0, A, B ∈ Σ , is said to be a ∗-submeasure. Similarly as in [1] we may state the question on existence of ∗-submeasure in terms of functional inequalities, more precisely to solve the following problem: to find a real-valued function f : [0, +∞[×[0, +∞[→ [0, 1] satisfying the following properties (a) f (0, x) = 1 for all x > 0; (b) f (a, x) is non-increasing in its first component; (c) f is a solution of the inequality f ( a + b, x) ≥

x

∫

f (a, x − t ) df (b, t ),

x > 0, a, b ≥ 0,

0

where the integral is meant in the sense of Riemann–Stieltjes. More generally, if C is a copula and L ∈ L, we define the function σL,C : ∆+ × ∆+ → ∆+ for x ∈ R+ by

σL,C (F , G)(x) =

 0,   ∫   

L(x)

x = 0, dC (F (u), G(v)),

1,

x ∈]0, +∞[, x = +∞,

for each F , G ∈ ∆ , where L(x) = {(u, v); u, v ∈ R+ , L(u, v) < x} and the integral is of Lebesgue–Stieltjes type. Then a mapping γ : Σ → ∆+ satisfying (PS1), (PS2) and +

(PS7) γA∪B (x) ≥



L(x)

dC (γA (u), γB (v)), x > 0, A, B ∈ Σ ,

is said to be a σL,C -submeasure. Acknowledgements This work was partially supported by grants VEGA 02/0097/08, APVV-0012-07 and MSM VZ 619889701. References [1] O. Hutník, R. Mesiar, On a certain class of submeasures based on triangular norms, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 17 (3) (2009) 297–316. [2] M. Grabiec, Y.L. Cho, V. Radu, On Nonsymmetric Topological and Probabilistic Structures, Nova Science Publ., New York, 2006. [3] B. De Baets, R. Mesiar, Metrics and T -equalities, J. Math. Anal. Appl. 267 (2002) 531–547. [4] B. De Baets, R. Mesiar, Pseudo-metrics and T -equivalences, J. Fuzzy Math. 5 (1997) 471–481. [5] D. Mihet, V. Radu, T -equalities as fuzzy non-Archimedean metrics and ∆+ -fuzzy equivalences of type S, Fuzzy Sets and Systems 157 (2006) 2751–2761. [6] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, in: North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing, New York, 1983. [7] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, in: Trends in Logic, Studia Logica Library, vol. 8, Kluwer Academic Publishers, 2000. [8] E.P. Klement, R. Mesiar (Eds.), Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, Elsevier, Amsterdam, 2005. [9] S. Saminger-Platz, C. Sempi, A primer on triangle functions I, Aequationes Math. 76 (2008) 201–240. [10] P.S. Mostert, A.L. Shields, On the structure of semigroups on a compact manifold with boundary, Ann. of Math. 65 (2) (1957) 117–143. [11] A. Kolesárová, A note on Archimedean triangular norms, BUSEFAL 80 (1999) 57–60. [12] T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators. New Trends and Applications, Physica-Verlag, Heidelberg, 2002. [13] C. Alsina, M.J. Frank, B. Schweizer, Associative Functions: Triangular Norms and Copulas, World Scientific Publishing Company, 2006.