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ON FUNCTIONS PRESERVING PRODUCTS OF

CERTAIN CLASSES OF SEMIMETRIC SPACES

MATEUSZ LICHMAN, PIOTR NOWAKOWSKI, AND FILIP TUROBOŚ

Abstract. In the paper we continue the research of Borsík andDoboš on functions which allow us to introduce a metric to theproduct of metric spaces. In this paper we extend their scope onbroader class of spaces which usually fail to satisfy the triangleinequality, albeit they tend to satisfy some weaker form of this ax-iom. In particular, we examine the behavior of functions preservingb-metric inequality. We provide analogues of the results of Borsíkand Doboš, adjusted to the new, broader setting. The results weobtained are illustrated with multitude of examples. Furthermore,the connections of newly introduced families of functions with theones already known from the literature are investigated.

1. Introduction

In the case of various structures of either algebraic type it is rela-tively easy to merge a finite number of such spaces into a single productspace equipped with structure based on the initial ones. When metricstructures are combined, usually functions like minimum, sum or qua-dratic mean are employed for this task. The investigation of functionswhich coalesce multiple metric spaces into a single one dates back tothe beginning of 1980s’, when Borsík and Doboš [4, 5] laid formal foun-dations to the concept of the product-wise metric preserving functions.The research on this concept was conducted in two different directions.Some researchers either sought more properties of such functions (e.g.[38, 39, 45]). Others considered a variation of this notion, motivatedby some applications, where a family of metrics were defined on a sameset and it was necessary to meld them into a single one. This approachresulted in the definition of aggregation functions – compare with theneat applicational papers of Valero et al., [31, 32, 33, 34] and see alsothe references therein.

The simultaneous development of the property preserving functionsof one variable is also worth mentioning. The idea can be traced back toWilson [51] and Sreenivasan [43]. The research on this subject was later

Date: December 2020.1

http://arxiv.org/abs/2012.15614v1

2 MATEUSZ LICHMAN, PIOTR NOWAKOWSKI, AND FILIP TUROBOŚ

conducted by Borsík, Doboš and Piotrowski (see [3, 4, 5, 6, 10, 11, 12,13, 14]), Corazza (see [7]), Das (see [9]), Dovgoshey (see [15, 16, 17, 49]),Jůza (see [25]), Khemaratchatakumthorn, Pongsriiam, Termwuttipongand Samphavat (see [27, 28, 29, 40, 41, 42, 44]), Pokorny (see [37,38, 39]), Vallin (see [47, 48, 49]) and recently also by Jachymski andTuroboś (see [22, 46]).

The concept of b-metric spaces can be traced back to Frink (see [18])as well as Bakhtin (see [1]), but the name is usually connected withCzerwik [8], who conducted the research on particular subclass of thisspaces where the relaxing constant was 2. Recently this generalizationof the notion of metric space is reliving its second youth, as the interestin this direction of metric-related research has been rekindled in recent30 years. This renaissance refers both to the advances in the field ofmetric fixed point theory as well as properties of these spaces them-selves (see [2, 21, 26, 50]). In the context of preserving triangle-likeconditions, a great deal of work has been done by aforementioned Thaimathematicians as well as Dovgoshey.

2. Preliminaries

We begin with introducing several important notions as well as someexamples depicting those. Throughout the paper, by R+ we will denotethe set of non-negative reals, i.e., R+ = [0,∞). Let us start with thenotion of semimetric.

Definition 1. Let X be a non-empty set and d : X2 → R+. We saythat d is a semimetric on X if the following conditions are satisfied:

(S1) ∀x,y∈X d(x, y) = 0 ⇐⇒ x = y;(S2) ∀x,y∈X d(x, y) = d(y, x).

Since the paper revolves around various generalizations of the tri-angle inequality, we start with the following two general definitionsinspired by the definition of triangle function introduced by Páles andBessenyei [2]:

Definition 2. A function f : R2+ → R+ is called nonreducing if, for

any a, b ∈ R2+, f(a, b) > max{a, b}.

Definition 3. Let g : R2+ → R+ be a nonreducing function. Let (X, d)

be a semimetric space satisfying the following condition in place of thetriangle condition:

(G) ∀x,y,z∈Xid(x, z) 6 g(d(x, y), d(y, z)).

Then we say that (X, d) is a G-metric space (and, accordingly, that dis a G-metric), where condition (G) is called a semi-triangle conditiongenerated by the function g.

ON FUNCTIONS PRESERVING PRODUCTS OF SEMIMETRIC SPACES 3

Example 4. We now give examples of some popular conditions withfunction g : R2

+ → R+ generating them (here K > 1 is some fixed,arbitrary constant):

(M) ∀x,y,z∈Xid(x, z) 6 d(x, y) + d(y, z), g(a, b) = a + b,

(U) ∀x,y,z∈Xid(x, z) 6 max{d(x, y), d(y, z)}, g(a, b) = max{a, b},

(BK) ∀x,y,z∈Xid(x, z) 6 K(d(x, y) + d(y, z)), g(a, b) = K(a + b).

(SK) ∀x,y,z∈Xid(x, z) 6 Kd(x, y) + d(y, z), g(a, b) = Ka + b,

(T) ∀x,y,z∈Xid(x, z) 6 ψ(d(x, y)+d(y, z)), g(a, b) = ψ(a+b), where

ψ : R+ → R+ is nondecresing, continuous at the origin andψ(0) = 0.

While the first two examples depict well-known notions of metric andultrametric respectively, third and fourth refer to the concepts knownas b-metric and strong b-metric spaces. We refer the Reader to theappropriate chapters of [30] for more examples and application of theseconcepts and we underline the fact that K is fixed in this situations,which will be important later on. Last point refers to the notion oftriangle function introduced in [2].

Let n ∈ N. For i ∈ {1, . . . , n}, let gi : R2+ → R+ be a nonreducing

function and Gi be the semi-triangle condition generated by the func-tion gi. Let (Xi, di) be Gi-metric space for i = 1, . . . , n. Moreover, leth : R2

+ → R+ be a nonreducing function and H , analogously, be thecondition generated by the function h.

By X =∏n

i=1Xi we shall denote the Carthesian product of spacesXi. We want to describe all functions F : Rn

+ → R+ for which the

function D : X2 → R+ given by a formula

(1) D(x,y) = F (d1(x1, y1), . . . , dn(xn, yn)),

for any1 x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ X is an H-metric on X.Thus, we arrive at the following definition.

Definition 5. We say that F is (G1, . . . , Gn) − H-preserving if forany collection of Gi-metric spaces (Xi, di), i = 1, . . . , n, the function Dgiven by (1) is an H-metric on X. The family of all (G1, . . . , Gn)−H-preserving functions will be denoted by P(G1,...,Gn)−H .

Remark 6. This definition is coherent with the notation used in [22]and references therein. For example, the class of all functions whichcombine a finite family of b-metrics (regardless of the relaxation con-stants on each space) into a single b-metric on a product space will be

1In general, throughout the paper the elements of the Carthesian product willbe denoted by boldface font.


called (n)-b-metric preserving functions and the family of all such func-tions will be denoted by P n

B. When different axioms are considered, forexample functions combining multiple b-metrics into a metric, a name(n)-b-metric-metric preserving functions will be used. Again, the classof functions having this property will be denoted by P n

BM . Analogously,we define families P n

SB, P nSM etc.

Of course, when we want to specify the relaxation constants K instarting or resulting spaces, we will refer to the notation (BK), (SK)and so forth.

Let us begin with the well-known example in order to acquaint theReader with the motivation for this paper.

Example 7. Consider a standard Euclidean metric de on Rn. It is,in fact, result of combining (according to formula (1)) a function F1 :Rn

+ → R given by

F1(x1, . . . , xn) :=

√

√

√

√

n∑

i=1

x2i

with n copies of standard metric (x, y) 7→ |x− y| defined on R.Replacing F1 with F2 : R

n+ → R+ defined as

F2(x1, . . . , xn) :=n∑

i=1

xi

yields so called taxicab metric.2 In turn using function F3 defined asmaximum of its arguments yields metric known as Chebyshev distanceor L∞ metric.

The following lemma is an easy consequence of the definition of fam-ilies P(G1,...,Gn)−H .

Lemma 2.1. Let n ∈ N. For i ∈ {1, . . . , n}, let gi, hi, g, h : R2+ → R+

be nonreducing functions and Gi, Hi, G,H be the semi-triangle condi-tions generated, respectively, by the functions gi, hi, g, h. Assume thatF : Rn

+ → R+ is (G1, . . . , Gn)−G-preserving. Then:

(i) if for all i ∈ {1, . . . , n} and every a, b ∈ R+ we have gi(a, b) >hi(a, b) (equivalently Hi ⇒ Gi), then F is (H1, . . . , Hn) − G-preserving;

(ii) if for all a, b ∈ R+ we have g(a, b) 6 h(a, b) (equivalently G ⇒H), then F is (G1, . . . , Gn)−H-preserving.

2Also known as Manhattan metric or simply L1 metric.


Remark 8. The family of all (n)-b-metric-metric preserving functionscan be written as the following intersection.

P nBM =

⋂

K1,...,Kn>1

P(BK1,BK2

,...,BKn )−M .

However, if two relaxed conditions are concerned, this class becomesan intersection of unions of particular families of functions (this factis an immediate consequence of Lemma 3.4, which will be proven inthe latter part of the paper). For example, when the class P n

B of all(n)-b-metric preserving functions is considered, the following equalityholds

P nB =

⋂

K1,...,Ki>1

⋃

K>1

P(BK1,BK2

,...,BKn )−BK.

Of course, analogous statements can be given for families connectedwith inequalities (S) and so on.

As one can see, the functions which are of the main concern in thispaper have multiple arguments. This require us to specify several no-tions, which are less obvious than their one-dimensional equivalents.

Definition 9. Let n ∈ N. We define a relation � on Rn+ by: a � b

iff ai 6 bi for all i ∈ {1, . . . , n} where a = (a1, . . . , an) and b =(b1, . . . , bn). We will also write a + b := (a1 + b1, . . . , an + bn) and0 := (0, . . . , 0).

Obviously, � is a partial order. The same partial order was consid-ered in the multiple papers of Valero et al. [31, 32] as well as in theoriginal works of Borsík and Doboš. In the light of the foregoing, weintroduce the definitions of the following properties:

Definition 10. Let n ∈ N. A function F : Rn+ → R+ is said to be

• monotone, if F (a) 6 F (b) whenever a � b;• subadditive, if F (a + b) 6 F (a) + F (b) for each a, b ∈ Rn

+;• quasi-subadditive, if there exists s > 1 such that F (a + b) 6

s(F (a) + F (b)) for each a, b ∈ Rn+.

Another crucial property of one-dimensional metric-preserving func-tions is known under the name amenability. A function f : R+ → R+

is called amenable if f−1 ({0}) = {0}. It seems reasonable to define itsmultidimensional equivalent as follows:

Definition 11. A function F : Rn+ → R+ is said to be amenable if the

following equivalence holds:

F (x) = 0 ⇐⇒ x = (0, 0, . . . , 0).


3. Main results

We’d like to start with a characterization of the (G1, . . . , Gn) − Hpreserving functions which is based on the concept of triangle triplet.The notion of the triangle triplet can be traced back to the paperof Sreenivasan [43] and was reintroduced in [3]. It is worth pointingout, that the usage of this concept for more general spaces appears in[27, 41] as well as in [46]. A successful attempt of extending the originaldefinition into multidimensional version have also been undertaken, see[4].

We will now unify all the concepts discussed above via the followingdefinition:

Definition 12. Let G be the semi-triangle condition generated by thenonreducing function g : R2

+ → R+. A triplet (a, b, c) ∈ [0,+∞)3 iscalled a G-triangle triplet if a 6 g(b, c), b 6 g(c, a), c 6 g(a, b) anda, b, c > 0 or (a, b, c) = (0, l, l) for some non-negative constant l, up toa permutation.

Remark 13. This specification in the definition above enables us toavoid the pitfall of triplets consisting of 0 and two distinct positivenumbers. Such triplet of distances could satisfy the desired inequalitiesfrom the Definition 12 despite being unobtainable in any semimetricspace due to the symmetry axiom. These triplets could cause addi-tional issues in formulation of some Theorems as well as in the proofs,therefore we have decided to exclude them from the Definition above.

We will use the symbol △G to denote the set of all G-triangle triplets(similar to the notions used in [42] as well as in their previous papers).

Definition 14. Let n ∈ N. For all i ∈ {1, . . . , n} let Gi be the semi-triangle condition generated by the nonreducing function gi : R

2+ →

R+. A triplet (a, b, c) ∈ [0,+∞)3n, where a, b, c ∈ Rn+ is called a

(G1, . . . , Gn)-triangle triplet if for each i = 1, . . . , n, the i-th coordi-nates of a, b and c form a Gi-triangle triplet (according to the Defini-tion 12).

Analogously to the one-dimensional case, we will denote the set ofall (G1, . . . , Gn)-triangle triplets by △(G1,...,Gn).

We can now move on to extending the results obtained in [4, 7, 14,42, 46] to a multidimensional scope.

Theorem 3.1. Let n ∈ N. For i ∈ {1, . . . , n}, let gi : R2+ → R+ be

a nonreducing function, Gi be the semi-triangle condition generated bythe function gi. Moreover, let h : R2

+ → R+ be a nonreducing function


and H be the semi-triangle condition generated by the function h. Afunction F : Rn

+ → R+ is (G1, . . . , Gn)−H-preserving if and only if itis amenable and satisfies the following condition

(2) ∀(a,b,c)∈△(G1,...,Gn)(F (a), F (b), F (c)) ∈ △H .

Proof. "⇐ " For i ∈ {1, . . . , n}, let (Xi, di) be a Gi-metric space.Without loss of generality, we may assume that these are non-trivial,i.e. consists of at least two points. Let X =

∏n

i=1Xi. Let x =(x1, . . . , xn),y = (y1, . . . , yn), z = (z1, . . . , zn) ∈ X. Since F is amenable,we have

D(x,y) = 0 ⇔ F (d1(x1, y1), . . . , dn(xn, yn)) = 0

⇔ ∀i∈{1,...,n} di(xi, yi) = 0

⇔ x = y.

By the definition of D and the fact that di are Gi-metrics, we have thatD is symmetric.

Since di are Gi-metrics, we have that, for any i ∈ {1, . . . , n},di(xi, zi) 6 gi(di(xi, yi), di(yi, zi)),

di(xi, yi) 6 gi(di(yi, zi), di(xi, zi)),

di(yi, zi) 6 gi(di(xi, zi), di(xi, yi)),

so (di(xi, zi), di(xi, yi), di(yi, zi)) forms a Gi-triangle triplet. Hence, by(2), we have that

D(x, z) = F (d1(x1, z1), . . . , dn(xn, zn))

6 h(F (d1(x1, y1), . . . , dn(xn, yn)), F (d1(y1, z1), . . . , dn(yn, zn)))

= h(D(x,y), D(y, z)).

Therefore, F is (G1, . . . , Gn)−H-metric preserving."⇒"Let F be (G1, . . . , Gn) −H-metric preserving. On the contrary, as-

sume that F is not amenable. If F (0) 6= 0, then a contradiction isobvious.

Thus, there is a = (a1, . . . , an) ∈ Rn+ \ {0} such that F (a) = 0.

For i ∈ {1, . . . , n} let (Xi, di) = ({0, ai}, de), where de is the Euclideanmetric. Then, for any i ∈ {1, . . . , n}, (Xi, di) is, obviously, a Gi-metricspace. Indeed, the first two axioms are trivial and the third followsfrom the fact that function gi is nonreducing for each i.

Let xi = 0, yi = ai, for i ∈ {1, . . . , n}, x = (x1, . . . , xn),y =(y1, . . . , yn). Then x 6= y and

D(x,y) = F (d1(x1, y1), . . . , dn(xn, yn)) = F (a1, . . . , an) = 0,


therefore we arrive at the contradiction.Now, assume on the contrary that condtion (2) is not satisfied. Then,

there exist a = (a1, . . . , an), b = (b1, . . . , bn), c = (c1, . . . , cn) ∈ Rn+

such that, for any i ∈ {1, . . . , n}, (ai, bi, ci) is a Gi-triangle triplet, butF (c) > h(F (a), F (b)). For i ∈ {1, . . . , n} define the space (Xi, di) inthe following way. Let Xi = {xi, yi, zi} (where xi, yi, zi are arbitrarybut not necessarily distinct points – in the case where some of thevalues below equal 0) and

di(xi, yi) = di(yi, xi) = ai,

di(yi, zi) = di(zi, yi) = bi,

di(xi, zi) = di(zi, xi) = ci,

di(xi, xi) = di(yi, yi) = di(zi, zi) = 0.

Then, for any i ∈ {1, . . . , n}, (Xi, di) is Gi-metric space. Indeed, thefirst two axioms hold trivially. The third one follows from the fact thatgi are nonreducing and the assumpion that (ai, bi, ci) is a Gi-triangletriplet. Let x = (x1, . . . , xn),y = (y1, . . . , yn), z = (z1, . . . , zn). Wehave

D(x, z) = F (d1(x1, z1), . . . , dn(xn, zn))

= F (c) > h(F (a), F (b))

= h(F (d1(x1, y1), . . . , dn(xn, yn)), F (d1(y1, z1), . . . , dn(yn, zn)))

= h(D(x,y), D(y, z)).

Thus, we arrive at another contradiction, which finishes the proof. �

Example 15. Let n ∈ N, F : Rn+ → R+ be given by the formula

F (a1, . . . , an) =a1 + · · ·+ an

n

and let K > 1. We will show that F is (BK , . . . , BK) − BK metricpreserving. F is obviously amenable. We need to prove that it satis-fies condition (2). For i ∈ {1, . . . , n} let ai, bi, ci ∈ R+ be such that(ai, bi, ci) ∈ △BK

for i = 1, 2, . . . , n. Then we have

F (c1, . . . , cn) =c1 + · · ·+ cn

n

6K(a1 + b1) + · · ·+K(an + bn)

n

= K · a1 + · · ·+ ann

+K · b1 + · · ·+ bnn

= K(F (a1, . . . , an) + F (b1, . . . , bn)).


Similarly, we prove that

F (a1, . . . , an) 6 K(F (c1, . . . , cn) + F (b1, . . . , bn))

andF (b1, . . . , bn) 6 K(F (c1, . . . , cn) + F (a1, . . . , an)).

Therefore, (F (a1, . . . , an), F (b1, . . . , bn), F (c1, . . . , cn)) ∈ △BK. By The-

orem 3.1, F is (BK , . . . , BK)−BK metric preserving. In particular (forK = 1), F is (n)-metric preserving.

While arithmetic mean well preserves the additional properties ofsemimetric spaces, this is not the case when geometric mean is consid-ered.

Example 16. Let F : Rn+ → R+ be given by the formula

F (a1, a2, . . . , an) = n√a1a2 . . . an.

Clearly, F is not amenable for n > 2. Due to Theorem 3.1, it cannotpreserve any semimetric properties.

The insightful Reader should notice already that if conditions Gi’sand H are the b-metric or strong b-metric inequalities, then the afore-mentioned theorem provides the characterization only for fixed valuesof relaxation constants K in the respective definitions of those inequal-ities. We shall now try to extend this characterization to all values ofK. Let us begin with the following remark.

Remark 17. For semimetric space (X, d) satisfying either (BK) or(SK) with relaxation constant K > 1 it is worth pointing out that itsatisfies the same condition for any K ′ > K. Indeed, for any threepoints x, y, z ∈ X we have (reasoning for condition (SK) is almostunchanged):

d(x, z) 6 K (d(x, y) + d(y, z)) 6 K ′ (d(x, y) + d(y, z)) .

We shall use this observation later on.

For subsequent results we will need two Lemmas of Turoboś [46,Lemma 3.1, 3.2] which allow us to combine multiple b-metric spacesinto a single one. These results are generalizations of well-known resultsfrom the theory of metric spaces.

Lemma 3.2. Let (X1, d1), (X2, d2) be a pair of disjoint, [strong] b-metric spaces with relaxation constants K1, K2 > 1. Additionally, as-sume that both spaces have finite diameter, i.e.,

r1 := diamd1(X1) = supx,y∈X1

d1(x, y) <∞


r2 := diamd2(X2) = supx,y∈X2

d2(x, y) <∞

and X := X1 ∪ X2 has at least three elements. Then, there existsa [strong] b-metric d : X × X → [0,+∞) with relaxation constantK := max{K1, K2}, diam(X) = max{r1, r2} and satisfying

d|X1×X1= d1 ∧ d|X2×X2

= d2.

Lemma 3.3. Let (Xn, dn)n∈N be an increasing family of [strong] b-metric spaces with a common, fixed relaxation constant K > 1 i.e. forevery pair of indices k1 6 k2 we have Xk1 ⊆ Xk2 and dk1 ⊆ dk2 (whichmeans dk1(x, y) = dk2(x, y) for all x, y ∈ Xk1). Then a pair (X, d),where

X :=⋃

n∈N

Xn, d(x, y) := dkmin(x, y)

and kmin is any index such that x, y ∈ Xkminis a [strong] b-metric space

with the relaxation constant K > 1.

Now, we are able to proceed with the following four-in-one lemma,describing the behaviour of (n)-b-metric preserving functions as wellas those which are (n)-strong-b-metric-b-metric, (n)-strong-b-metric or(n)-b-metric-strong-b-metric preserving. As the proof in all cases fol-lows the same train of thought, we present only one of its variants.

Lemma 3.4. Let n ∈ N. Assume that F is a function such that forevery n-element collection of [strong] b-metric spaces (Xi, di) with fixedrelaxation constants Ki, i = 1, . . . , n, the product space (X, D) – whereX :=

∏n

i=1Xi and D is given by (1) – is a [strong] b-metric space.Then, there exists K ′ > 1 such that:

a) for every [strong] (BK1, . . . , BKn)-triangle triplets a, b, c, the

values f(a), f(b) and f(c) form a [strong] BK ′-triangle triplet.b) the relaxation constant of the resulting space (X, D) is bounded

by K ′.

Proof. Due to the fact, that each version of this lemma is proved bythe exact same method, we will simply prove the version concerningb-metric spaces.

Let f be a function satisfying the assumptions of our Lemma. Forthe sake of convenience, let us denote K := (K1, . . . , Kn).

Let us suppose that the contrary to a) holds. In particular, it impliesthat for every k ∈ N there exists a (BK1, . . . , BKn

)-triangle triplet3

3Throughout the paper for denoting vectors coordinates we use subscript in-dexation. Thus, the sequence of vectors

(

a(n))

n∈Nwill be marked by their upper

indexation. For example, 4-th coordinate of a(6) will be denoted by a(6)4 .


(a(k), b(k), c(k)) for which(

f(a(k)), f(b(k)), f(c(k)))

fails to be a one-

dimensional Bk-triangle triplet.Without loss of generality, we will assume that

(3) ∀k∈N F (a(k)) > k ·(

F (b(k)) + F (c(k)))

,

but this additional assumption on the order of elements appearing inthe discussed inequality serves only increasing the clarity of the proof.

Fix j ∈ {1, . . . , n}. Let X̂(1)j = X

(1)j := {(1, j, 1), (1, j, 2), (1, j, 3)}

and define b-metric d(1)j on this set as follows:

d(1)j ((1, j, 1), (1, j, 2)) = a

(1)j

d(1)j ((1, j, 2), (1, j, 3)) = b

(1)j ,

d(1)j ((1, j, 1), (1, j, 3)) = c

(1)j .

Remaining values of function d(1)j are the result of semimetric axioms.

An easy observation is that when a(1)j , b

(1)j , c

(1)j are non-zero, then d

(1)j

is a b-metric with relaxation constant Kj (this fact is an immediateconsequence of the Definition 14).

However, if one or more of these values (for example a(1)j ) equals zero,

then we can consider spaces consisting of less than three points simplyby glueing some of those points together. As the Reader will see inthe latter part of the proof, this identification of two or three pointsas a single one for these situations holds for the rest of the reasoning.Therefore, we will not focus on these special cases and for the sake of

clarity we will assume that a(k)j , b

(k)j , c

(k)j > 0.

Now, assume that we have defined(

X̂(k−1)j , d̂

(k−1)j

)

, which is a finite

(thus bounded) b-metric space with relaxation constant Kj. We now

proceed with defining X̂(k). Put X(k)j := {(k, j, 1), (k, j, 2), (k, j, 3)}

and d(k)j : X

(k)j ×X

(k)j → [0,+∞), as a unique semimetric satisfying

d(k)j ((k, j, 1), (k, j, 2)) = a

(k)j ,

d(k)j ((k, j, 2), (k, j, 3)) = b

(k)j ,

d(k)j ((k, j, 1), (k, j, 3)) = c

(k)j .

The space (X(k)j , d

(k)j ) is a bounded b-metric space with relaxation con-

stant Kj . We can now use Lemma 3.2, to obtain b-metric space with

same relaxation constant, defined on a set X̂(k)j := X

(k)j ∪ X̂(k−1)

j . The


Lemma guarantees that the extension d̂(k)j equals d

(k)j on X

(k)j × X

(k)j

and d̂(k−1)j on X̂

(k−1)j × X̂

(k−1)j .

By this step-wise procedure, we obtain a sequence of b-metric spaces((

X̂(k)j , d̂

(k)j

))

n∈N. Due to Lemma 3.3, we thus obtain a b-metric space

(Xj, Dj) where Xj :=⋃

k∈NX(k)j and

Dj(x, y) := d̂(kx,y)j (x, y) where kx,y := min{k ∈ N : x, y ∈ X

(k)j }.

Repeating this procedure for each j we obtain a collection of n b-metric spaces with relaxation constants K1, . . . , Kn respectively. Define(X, D) as in the (1), that is:

X :=

n∏

j=1

Xj , D(x,y) := F (D1(x1, y1), . . . , Dn(xn, yn)) ,

where x = (x1, . . . , xn) ∈ X, y likewise. To prove our claim, we simplyneed to ensure ourselves that (X , D) fails to be a b-metric space.

Suppose otherwise, i.e., (BK ′) inequality holds in (X, D) for someK ′ > 1. Consider any k0 > K ′. By Remark 17 if b-metric inequalitywith constant K ′ holds, then condition (Bk0) holds as well. Considerelements x,y, z ∈ X of the form

x := ((k0, 1, 1), (k0, 2, 1), . . . , (k0, n, 1));

y := ((k0, 1, 2), (k0, 2, 2), . . . , (k0, n, 2));

z := ((k0, 1, 3), (k0, 2, 3), . . . , (k0, n, 3)).

Notice that for each j the inequality

Dj((k0, j, 1), (k0, j, 2)) = a(k0)j 6 Kj · (b(k0)j + c

(k0)j )

=Kj · (Dj((k0, j, 2), (k0, j, 3)) +Dj((k0, j, 3), (k0, j, 1)))

holds. However, from the assumption (3) we obtain that

D(x,y) = F (a(k0)) > k0 ·(

F (b(k0)) + F (c(k0)))

= k0 · (D(x, z) +D(z,y)) .

This proves, that D fails to satisfy (Bk0) and, by Remark 17, it also failsto do so with condition (BK ′). Due to K ′ being arbitrary, the semi-metric space (X, D) fails to be a b-metric space at all, a contradiction.Thus F satisfies a). The condition b) is a straightforward consequenceof a). �

Corollary 3.5. If F ∈ P nB then for all K1, . . . , Kn > 1 we have that

F ∈ P(BK1,...,BKn)−BK

for some K > 1.


Remark 18. From Lemma 3.4 and Theorem 3.1 it is possible toobtain an (n)-dimensional generalization of Theorem 4.1 from [46].Among the others, it states that F ∈ P n

MB if and only if there ex-ists K > 1 such that for any (M, . . . ,M)-triangle triplet (a, b, c), thevalues (F (a), F (b), F (c)) form a BK-triangle triplet. Similar charac-terizations can be given for other classes like P n

BS, P nS and so forth.

The conditions above are, in general, hard to verify. Therefore a needfor more convenient conditions arises. We begin with the following,relatably easy to check sufficient condition.

Proposition 3.6. Let n ∈ N and F : Rn+ → R+ be a function. If F is

amenable, monotone and quasi-subadditive with a constant s > 1, thenF is (n)-b-metric preserving.

Proof. Let {(X1, d1), . . . , (Xn, dn)} be a collection of b-metric spaceswith constants K1, . . . , Kn respectively. Let X := X1 × · · · × Xn.Consider a function D : X ×X → R+ given by

D(x,y) = F (d1(x1, y1), . . . , dn(xn, yn))

for each x,y ∈ X. Since F is amenable, D(x,y) = 0 if and only ifd1(x1, y1) = · · · = dn(xn, yn) = 0, which is equivalent to (as d1, . . . , dnare all b-metrics) x = y. The symmetry of D clearly comes from thesymmetry of d1, . . . , dn.

Lastly, we shall prove that D is a b-metric. Let K ∈ N be such thatK > max{K1, . . . , Kn}. Put

s′ :=sK+1 − s2

s− 1+ sK .

Let x,y, z ∈ X and observe that from the quasi-subadditivity of F itfollows that for any natural N > 2 and any (a1, . . . , an) ∈ Rn

+, we have

F (Na1, . . . , Nan) = F ((N − 1)a1 + a1, . . . , (N − 1)an + an)

6 sF ((N − 1)a1, . . . , (N − 1)an) + sF (a1, . . . , an).

If N > 3, then we can repeat this procedure for N − 1, obtaining

F (Na1, . . . , Nan) 6 s2F ((N − 2)a1, . . . , (N − 2)an)

+s2F (a1, . . . , an) + sF (a1, . . . , an).


An easy induction leads us to the following inequality

F (Na1, . . . , Nan) 6 sN−1F (a1, . . . , an) + sN−1F (a1, . . . , an)

+sN−2F (a1, . . . , an) + · · ·+ sF (a1, . . . , an)

=

(

N−1∑

i=1

si + sN−1

)

F (a1, . . . , an)

=

(

sN − s

s− 1+ sN−1

)

F (a1, . . . , an).

Using the above inequality, the monotonocity of F and the fact thatdi(xi, yi) 6 K(di(xi, zi) + di(zi, yi)), for i = 1, . . . , n, we have

D(x,y) = F (d1(x1, y1), . . . , dn(xn, yn))

6 F (K(d1(x1, z1) + d1(z1, y1)), . . . ,K(dn(xn, zn) + dn(zn, yn)))

6

(

sK − s

s− 1+ sK−1

)

F (d1(x1, z1) + d1(z1, y1), . . . , dn(xn, zn) + dn(zn, yn))

6 s ·(

sK − s

s− 1+ sK−1

)

[F (d1(x1, z1), . . . , dn(xn, zn)) + F (d1(z1, y1), . . . , dn(zn, yn))]

6 s′(D(x,z) +D(z,y))

Thus, D is a b-metric with a constant s′. �

Proposition 3.7. Let n ∈ N and F : Rn+ → R+ be an amenable func-

tion. If there exist constants a, c ∈ R with 0 < a 6 c such that

∀x∈Rn+(x 6= (0, . . . , 0) =⇒ F (x) ∈ [a, c])

then F is (n)-b-metric preserving with a constant K = max{1, c2a}.

Proof. Let (X1, d1), . . . , (Xn, dn) be a collection of b-metric spaces. LetX := X1 × · · · ×Xn. Consider a function D : X ×X → R+ given by

D(x,y) = F (d1(x1, y1), . . . , dn(xn, yn))

for each x,y ∈ X. In similar fashion to the proof of Proposition 3.6we can verify that D fulfills conditions (S1), (S2). Let x,y, z ∈ X

and K = max{1, c2a}. Without loss of generality, we can assume that

x,y, z are distinct. Then

a 6 F (d1(x1, z1), ..., dn(xn, zn))

as well as

a 6 F (d1(z1, y1), ..., dn(zn, yn)).


In particular, this implies that

D(x,y) = F (d1(x1, y1), ..., dn(xn, yn)) 6 c 6 K · 2a6 K(F (d1(x1, z1), ..., dn(xn, zn)) + F (d1(z1, y1), ..., dn(zn, yn)))

= K(D(x, z) +D(z,y)),

which concludes the proof. �

Corollary 3.8. Let n ∈ N and F : Rn+ → R+. If F is amenable and

there exists c > 0 such that F (a) ∈ [c, 2c] for all a ∈ Rn+ \ {0}, then

F ∈ P nBM .

In one-dimensional variant, the implication in the Corollary abovecan be reversed, as shown in [27, Theorem 24]. This is not the casewhen n > 2 is concerned. The proper counterexample is presentedbelow.

Example 19. Let F : R2+ → R+ be given by a formula

F (a) =

0 if a = (0, 0)1 if a = (1, 0)3 if a = (0, 1)2 otherwise.

We will show that F ∈ P 2BM despite not satisfying the condition from

Corollary 3.8. Let a = (a1, a2), b = (b1, b2), c = (c1, c2) ∈ R2+ \ {(0, 0)}.

Assume that (F (a), F (b), F (c)) /∈ ∆M . By Theorem 3.1, it sufficesto show that for i = 1 or i = 2 we have that (ai, bi, ci) /∈ ∆BK

forany K > 1. Since a, b, c 6= (0, 0), we have that one of a, b, c (say c)must be equal to (0, 1) and a = b = (1, 0). Then F (c) = 3 > 1 + 1 =F (a) + F (b). However c2 > K(a2 + b2) = 0 for any K > 1. Therefore,(a, b, c) /∈ ∆BK

. Finally, F ∈ P 2BM .

Example 20. Let n ∈ N, F : Rn+ → R+ be the sum of squares of its

arguments, i.e.

F (a1, . . . , an) =

n∑

i=1

a2i .

F is clearly monotone and amenable. We will show that it is quasi-subadditive. Let a1, . . . , an, b1, . . . , bn ∈ R+. Then

F (a1 + b1, . . . , an + bn) =n∑

i=1

(ai + bi)2 =

n∑

i=1

(

a2i + 2aibi + b2i)

6

n∑

i=1

(

2a2i + 2b2i)

=n∑

i=1

2a2i +n∑

i=1

2b2i

= 2F (a1, . . . , an) + 2F (b1, . . . , bn).


Hence F is quasi-subbadditive and, by Proposition 3.6, it is (n)-b-metric preserving. Observe that F is not (n)-metric preserving. Indeed,consider the triple (1, 2, 3). Of course, (1, 2, 3) ∈ △M . However,

F (3, . . . , 3) = 9n > 5n = n+ 4n = F (1, . . . , 1) + F (2, . . . , 2).

So, (F (1, . . . , 1), F (2, . . . , 2), F (3, . . . , 3)) /∈ △M . By Theorem 3.1, F isnot (n)-metric preserving.

The above example proves that conditions from Proposition 3.7 arenot necessary. On the other hand, using Proposition 3.7 one can easilyconstruct an example of an (n)-b-metric preserving function indicatingthat the condition of monotonicity from Proposition 3.6 is also notnecessary.

Example 21. Let n ∈ N, F : Rn+ → R+ be given by the formula

F (x1, ..., xn) =

0 if x1 = ... = xn = 0

2 if x1 = ... = xn = 1

1 otherwise

In the light of Proposition 3.7, F is (n)-b-metric preserving. Clearly, itis not monotone.

As the Reader can see, the monotonicity is not necessary for a func-tion to be (n)-b-metric preserving. Theorem 3.1 states that amenabilitycannot be omitted and the following results show that neither can bequasi-subadditivity.

Lemma 3.9. Let n ∈ N and F : Rn+ → R+, F ∈ P n

MB. Then F isquasi-subadditive.

Proof. By Remark 8, there exists a constant K > 1 such that F is(M, . . . ,M)− BK preserving.

Let a = (a1, . . . , an), b = (b1, . . . , bn) ∈ Rn+. It is easy to see that

(ai, bi, ai + bi) ∈ △M for i ∈ {1, . . . , n}. Therefore, by Theorem 3.1,

F (a + b) 6 K(F (a) + F (b))

for every a, b ∈ Rn+. Hence F is quasi-subadditive with a constant

K. �

Despite being useful on its own, we can use this Lemma to prove thefollowing

Theorem 3.10. Let n ∈ N and F : Rn+ → R+. Then F ∈ P n

B if andonly if F ∈ P n

MB.


Proof. The necessity is obvious. Now, let F ∈ P nMB and consider a

collection of b-metric spaces (X1, d1), . . . , (Xn, dn) with relaxation con-stants K1, . . . , Kn > 1 respectively. Let X := X1× · · ·×Xn. Considera function D : X ×X → R+ given by

D(x,y) = F (d1(x1, y1), . . . , dn(xn, yn))

for each x = (x1, . . . , xn),y = (y1, . . . , yn) ∈ X. Condition (S1) holdsfor the function D since F is amenable. (S2) is also obvious.

Using Lemma 3.9 we obtain the existence of s > 1 such that

(4) F (a + b) 6 s(F (a) + F (b)) for all a, b ∈ Rn+.

Since d1, . . . , dn are b-metrics, there existsN ∈ N,N > max{K1, . . . , Kn}such that for any i ∈ {1, . . . , n} and for all xi, yi, zi ∈ Xi

(5) di(xi, yi) 6 N (di(xi, zi) + di(zi, yi)) .

Using Theorem 3.1, Lemma 3.4 and the fact that F ∈ P nMB, we are

able to find K > 1 such that for any a = (a1, . . . , an), b = (b1, . . . , bn),c = (c1, . . . , cn) ∈ Rn

+ satisfying (ai, bi, ci) ∈ △M for i ∈ {1, . . . , n} wehave (F (a), F (b), F (c)) ∈ △BK

.

LetM := 2K

(

N∑

i=2

si + sN)

. Consider x = (x1, . . . , xn), y = (y1, . . . , yn),

z = (z1, . . . , zn) ∈ X and define a, b, c as follows:

a := (d1(x1, y1), . . . , dn(xn, yn)),

b := (d1(x1, z1), . . . , dn(xn, zn)),

c := (d1(y1, z1), . . . , dn(yn, zn)).

Then, by (5), we get

(6) ai 6 Nbi +Nci

for all i ∈ {1, . . . , n}. Therefore, (ai, Nbi + Nci, Nbi + Nci) ∈ △M fori ∈ {1, . . . , n}, and hence (F (a), F (Nb+ Nc), F (Nb+ Nc)) ∈ △BK

.Thus,

D(x,y) = F (a) 6 K(F (Nb+Nc) + F (Nb+Nc))

= 2K · F (N(b+ c)).

Analogously to the reasoning in the proof of Proposition 3.6 we canshow that for all m ∈ N

F (m · x) 6(

m−1∑

i=1

si + sm−1

)

F (x) for all x ∈ Rn+.


Combining (4) with the two inequalities above we obtain

D(x,y) 6 2K · F (N(b+c)) 6 2K

(

N−1∑

i=1

si + sN−1

)

F (b+c)

6 2K

(

N∑

i=2

si + sN

)

(F (b) + F (c))

= M(D(x, z) +D(z,y)),

which proves that D is a b-metric on X with a constant M . �

Example 22. Consider F : R2+ → R+ given by the formula

F (a, b) = ea+b for all (a, b) ∈ R2+.

We will show that F is not quasi-subadditive. Observe that

lima1→∞

lima2→∞

F (a1 + a2, 0)

F (a1, 0) + F (a2, 0)= lim

a1→∞lim

a2→∞

ea1+a2

ea1 + ea2

= lima1→∞

ea1 = ∞.

Hence for any s > 1 there exist a1, a2 ∈ R+ such that

F (a1 + a2, 0) > s(F (a1, 0) + F (a2, 0)).

Consequently, F is not quasi-subadditive and, by Lemma 3.9, it is not(2)-metric-b-metric preserving. By Theorem 3.10, F is not (2)-b-metricpreserving.

Proposition 3.11. Let n ∈ N. Then P nBM ( P n

M ( P nMB = P n

B.

Proof. The inclusions P nBM ⊂ P n

M and P nM ⊂ P n

MB follows from Lemma2.1 and Remark 8.

The equality P nMB = P n

B is precisely the statement of Theorem 3.10.The arithmetic mean F from Example 15 is (n)-metric preserving

but it does not convert the products of b-metrics to metrics. Indeed,taking the (B2, . . . , B2)-triplet (x,y, z) consisting of

x := (1, . . . , 1), y := (2, . . . , 2), z := (6, . . . , 6),

we obtain that F (x) = 1, F (y) = 2 and F (z) = 6. Since 6 > 2 + 1,these values do not form an M-triplet. Therefore from Theorem 3.1 itfollows that F /∈ P n

BM . Hence the inclusion P nBM ⊂ P n

M is proper aswell.

Lastly, the Example 20 shows that the inclusion P nM ⊂ P n

MB is alsoproper. �


4. Applications

It is worth pointing out that functions which combine multiple dis-tance functions on a product of metric-type spaces find their applica-tions in multiple-criteria decision making (MCDM). In particular, wewould like to discuss the application of property-preserving functionsregarding the TOPSIS method (introduced by Hwang and Yoon in 1981[20] and subsequently developed afterwards, e.g. [19, 23, 24, 36]). TOP-SIS is an acronym for the Technique for Order Preference by Similarityto the Ideal Solution. As the name suggests, the procedure ranks thealternatives according to two distances: the one from hypotheticallyideal solution and the one from theoretically worst alternative.

The TOPSIS algorithm starts with forming the decision matrix whichis meant to represent the satisfaction coming from the choice of each al-ternative according to the given criterion. Then, the matrix is normal-ized according to some normalization procedure. During this process,the values are multiplied by the criteria weights (the process of ob-taining these weights will not be discussed in this paper). Afterwards,the positive-ideal and negative-ideal solutions (often abbreviated byPIS and NIS) are constructed (usually by taking maximal and minimalpossible values for each criterion). Then, the distances of each availablealternative to PIS and NIS are calculated with a proper distance mea-sure, which stems from applying the product-wise property-preservingfunction (usually the metric-preserving variant, see e.g. [35]). At last,the alternatives are ranked based on the ratios of their distances fromthe negative-ideal solution to the sum of distances from the positive-ideal and the negative-ideal solutions. Of course, the higher the ratio,the better the alternative.

While some initial research has been done in this field, the effects ofchoice of function used to generate the metric for measuring distancesfrom PIS and NIS remain unbeknownst to a large extent, especiallywhen more general classes of semimetrics are discussed. While MCDMseems to be out of our area of expertise for a moment, we stronglyunderline that it remains an important field of applications for func-tions preserving certain classes of semimetric spaces discussed withinthe scope of this article.

5. Conclusion

Although the topic of generalizations of function preserving metric-type properties have not been investigated well in the multidimensionalcase, they bear strong similiraties to their one-dimensional equivalents.Investigation of those seems more important than their one-dimensional


equivalents though, as their field of application seems much broaderthan initially thought by mathematicians, involving multiple practicalaspects of computer studies such as database management, aforemen-tioned MCDM and so forth.

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Institute of Mathematics, Lodz University of Technology

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http://arxiv.org/abs/2011.14110

http://arxiv.org/abs/1912.10411

on functions preserving products of arxiv:2012.15614v1

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