On monoids of metric preserving functions

Let X be a class of metric spaces and let P_X be the set of all f : [0, ∞) → [0, ∞) preserving X, i.e., (Y, f ∘ ρ) ∈ X whenever (Y, ρ) ∈ X. For arbitrary subset A of the set of all metric preserving functions, we show that the equality P_X = A has a solution if A is a monoid with respect to the operation of function composition. In particular, for the set SI of all amenable subadditive increasing functions, there is a class X of metric spaces such that P_X = SI holds.

2020 Mathematics Subject Classification: Primary 26A30, Secondary 54E35, 20M20

1 Introduction

The following is a particular case of the concept introduced by Jachymski and Turoboś [1].

Definition 1. Let A be a class of metric spaces. Let us denote by P_A the set of all functions f : [0, ∞) → [0, ∞) such that the implication

$\begin{array}{l}((X,d)\in \text{A})\Rightarrow ((X,f\circ d)\in \text{A})\end{array}$

is valid for every metric space (X, d).

For mappings F : X → Y and Φ : Y → Z, we use the symbol F ∘ Φ to denote the mapping

$X\stackrel{F}{\to }Y\stackrel{\Phi }{\to }Z.$

We also use the following notation:

F, set of functions f : [0, ∞) → [0, ∞);

F₀, set of functions f ∈ F with f(0) = 0;

Am, set of functions f ∈ F₀ with f⁻¹(0) = {0};

SI, set of subadditive increasing f ∈ Am;

M, class of metric spaces;

U, class of ultrametric spaces;

Dis, class of discrete metric spaces;

M₂, class of two-points metric spaces;

M₁, class of one-point metric spaces.

The main purpose of this article is to provide a solution to the following problems.

Problem 2. Let A ⊆ P_M. Find conditions under which the equation

$\begin{array}{ll}{\text{P}}_{\text{X}}=\text{A}& (1)\end{array}$

has a solution X ⊆ M.

Problem 3. Let A ⊆ P_U. Find conditions under which Equation (1) has a solution X ⊆ U.

In addition, we find all solutions to Equation (1) for A equal to F, F₀, or Am and answer the following question.

Question 4. Is there a subclass X of the class M such that

${\text{P}}_{\text{X}}=\text{S}\text{I}?$

This question was posed as a challenge in [2] in a different but equivalent form and it was the original motivation for our research.

The article is organized as follows. The next section contains some necessary definitions and facts from the theories of metric spaces and metric preserving functions.

Section 3 provides some definitions from the semigroup theory and describes solutions to Equation (1), for the cases when A is F, F₀, or Am. In addition, we show that P_X is always a submonoid of (F, ∘). See Theorems 21, 23, 24, and Proposition 27, respectively.

Section 4 provides solutions to Problems 2 and 3, which are given, respectively, in Theorems 30 and 33. Theorem 32 gives a positive answer to Question 4.

2 Preliminaries on metrics and metric preserving functions

Let X be non-empty set. A function d : X × X → [0, ∞) is said to be a metric on the set X if for all x, y, z ∈ X we have

(i) d(x, y) ⩾ 0 with equality if and only if x = y, the positivity property;

(ii) d(x, y) = d(y, x), the symmetry property;

(iii) d(x, y) ⩽ d(x, z) + d(z, y), the triangle inequality.

A metric space (X, d) is ultrametric if the strong triangle inequality

$d(x,y)⩽max\left\{d(x,z),d(z,y)\right\}$

holds for all x, y, z ∈ X.

Example 5. Let us denote ${ℝ}_0^{+}$ by the set (0, ∞). Then the mapping $d^{+}:{ℝ}_0^{+}\times {ℝ}_0^{+}\to [0,\infty ),$

$d^{+}(p,q):=\{\begin{array}{ll}0& \text{if }p=q,\\ max\left\{p,q\right\}& \text{otherwise}.\end{array}$

is the ultrametric on ${ℝ}_0^{+}$ introduced by Delhommé et al. [3].

Definition 6. Let (X, d) be a metric space. The metric d is discrete if there is k ∈ (0, ∞) such that

$\begin{array}{l}d(x,y)=k\end{array}$

for all distinct x, y ∈ X.

In what follows we will say that a metric space (X, d) is discrete if d is a discrete metric on X. We will denote the class of all discrete metric space by Dis. In addition, for given non-empty set X₁, we will denote by Dis_X1 the subclass of Dis consisting of all metric spaces (X₁, d) with discrete d.

Remark 7. All discrete topological spaces can be endowed with a metric which is discrete, but not every metric space with discrete topology is discrete in the sense of Definition 6.

Example 8. Let M_k, for k = 1, 2, be the class of all metric spaces (X, d) satisfying the equality

$card(X)=k.$

Then all metric spaces belonging to M₁ ∪ M₂ are discrete.

Proposition 9. The following statements are equivalent for each metric space (X, d) ∈ M.

(i) (X, d) is discrete.

(ii) Every three-points subspace of (X, d) is discrete.

Proof. The implication (i) ⇒ (ii) is evidently valid.

Suppose that (ii) holds but (X, d) ∉ Dis. Then there are some different points i, j, k, l ∈ X such that

$\begin{array}{ll}d(i,j)\ne d(k,l).& (2)\end{array}$

We write X₁ : = {i, j, k} and X₂ : = {j, k, l}. Then the spaces (X₁, d|_X1×X1) and (X₂, d|_X2×X2) are discrete subspaces of (X, d) by statement (ii). Consequently we have

$\begin{array}{ll}d(i,j)=d(j,k)& (3)\end{array}$

and

$\begin{array}{ll}d(j,k)=d(k,l),& (4)\end{array}$

by definition of the class Dis. Now (3) and (4) give us

$d(i,j)=d(k,l),$

which contradicts (2).

Remark 10. The standard definition of discrete metric can be formulated as follows: “The metric on X is discrete if the distance from each point of X to every other point of X is one.” (see, for example, Searcóid [4]).

Let F be the set of all functions f : [0, ∞) → [0, ∞).

Definition 11. A function f ∈ F is metric preserving (ultrametric preserving) if f ∈ P_M (f ∈ P_U).

Remark 12. The concept of metric preserving functions can be traced back to Wilson [5]. Similar problems were considered by Blumenthal [6]. The theory of metric preserving functions was developed by Borsík, Doboš, Piotrowski, Vallin, and other mathematicians [7–19]. See also lectures by Doboš [20] and the introductory paper by Corazza [21]. The study of ultrametric preserving functions began by Pongsriiam and Termwuttipong [22] and was continued in [23, 24].

We will say that f ∈ F is amenable if

$\begin{array}{l}{f}^{-1}(0)=\left\{0\right\}\end{array}$

holds and the set of all amenable functions from F will be denoted by Am. Let us denote the set of all functions f ∈ F satisfying the equality f(0) = 0 by F₀. It follows directly from the definition that Am⊊F₀ ⊊ F.

Moreover, a function f ∈ F is increasing if the implication

$(x⩽y)\Rightarrow (f(x)⩽f(y))$

is valid for all x, y ∈ [0, ∞).

The following theorem was proved in [22].

Theorem 13. A function f ∈ F is ultrametric preserving if and only if f is increasing and amenable.

Remark 14. Theorem 13 was generalized in [25] to the special case of the so-called ultrametric distances. These distances were introduced by Priess-Crampe and Ribenboim [26] and were studies by different researchers [27–30].

Recall that a function f ∈ F is said to be subadditive if

$f(x+y)⩽f(x)+f(y)$

holds for all x, y ∈ [0, ∞). Let us denote the set of all subadditive increasing functions f ∈ Am by SI.

In the next proposition, we restate the equivalence between statements (i) and (ii) of Corollary 36 [2].

Proposition 15. The equality

$\begin{array}{l}\text{S}\text{I}={\text{P}}_{\text{U}}\cap {\text{P}}_{\text{M}}\end{array}$

holds.

Remark 16. The metric preserving functions can be considered as a special case of metric products (= metric preserving functions of several variables). See, for example, [31–37]. An important special class of ultrametric preserving functions of two variables was first considered in 2009 [38].

3 Preliminaries on semigroups. Solutions to F_X = A for A = F, F₀, and Am

Let us recall some basic concepts of semigroup theory, see, for example, “Fundamentals of Semigroup Theory” by Howie [39].

A semigroup is a pair (S, *) consisting of a non-empty set S and an associative operation * : S × S → S, which is called the multiplication on S. A semigroup S = (S, *) is a monoid if there is e ∈ S such that

$e*s=s*e=s$

for every s ∈ S.

Definition 17. Let (S, *) be a semigroup and ∅ ≠ T ⊆ S. Then T is a subsemigroup of S if a, b ∈ T ⇒ a * b ∈ T. If (S, *) is a monoid with the identity e, then T is a submonoid of S if T is a subsemigroup of S and e ∈ T.

Example 18. The semigroups (F, ∘), (Am, ∘), (P_M, ∘), and (P_U, ∘) are monoids, and the identical mapping id:[0, ∞) → [0, ∞), id(x) = x for every x ∈ [0, ∞) is the identity of these monoids.

The following simple lemmas are well-known.

Lemma 19. Let T be a submonoid of a monoid (S, *) and let V ⊆ T. Then V is a submonoid of (S, *) if and only if V is a submonoid of T.

Lemma 20. Let T₁ and T₂ be submonoids of a monoid (S, *). Then the intersection T₁ ∩ T₂ also is a submonoid of (S, *).

The next theorem describes all solutions to the equation P_X = F.

Theorem 21. The following statements are equivalent for every X ⊆ M.

(i) X is the empty subclass of M.

(ii) The equality

$\begin{array}{ll}{\text{P}}_{\text{X}}=\text{F}& (5)\end{array}$

holds.

Proof. (i) ⇒ (ii). Let X be the empty subclass of M. Definition 1 implies the inclusion F⊇P_X. Let us consider an arbitrary f ∈ F. To prove equality (5), it is suffice to show that f ∈ P_X. Since X is empty, the membership relation (X, d) ∈ X is false for every metric space (X, d). Consequently, the implication

$((X,d)\in \text{X})\Rightarrow ((X,f\circ d)\in \text{X})$

is valid for every (X, d) ∈ M. It implies f ∈ P_X by Definition 1. Equality (5) follows.

(ii) ⇒ (i). Let (ii) hold. We must show that X is empty. Suppose contrary that there is a metric space (X, d) ∈ X. Since, by definition, we have X ≠ ∅, there is a point x₀ ∈ X. Consequently, d(x₀, x₀) = 0 holds. Let c ∈ (0, ∞) and let f : [0, ∞) → [0, ∞) be a constant function,

$\begin{array}{l}f(t)=c\end{array}$

for every t ∈ [0, ∞). In particular, we have

$\begin{array}{ll}f(0)=c>0.& (6)\end{array}$

Equality (5) implies that f ∘ d is a metric on X. Thus, we have

$0=f(d(x_0,x_0))=f(0),$

which contradicts (6). Statement (i) follows.

Remark 22. Theorem 21 becomes invalid if we allow the empty metric space to be considered. The equality

${\text{P}}_{\text{X}}=\text{F}$

holds if the non-empty class X contains only the empty metric space.

Let us describe now all possible solutions to P_X = F₀.

Theorem 23. The equality

$\begin{array}{ll}{\text{P}}_{\text{X}}={\text{F}}_0& (7)\end{array}$

holds if and only if X is a non-empty subclass of M₁.

Proof. Let X ⊆ M₁ be non-empty. Equality (7) holds if

$\begin{array}{ll}{\text{P}}_{\text{X}}\supseteq {\text{F}}_0& (8)\end{array}$

and

$\begin{array}{ll}{\text{P}}_{\text{X}}\subseteq {\text{F}}_0.& (9)\end{array}$

Here, we prove the validity of (8). Let f ∈ F₀ be arbitrary. Since every (X, d) ∈ X is a one-point metric space, we have f ∘ d = d for all (X, d) ∈ X by the positivity property of metric spaces, Inclusion (8) follows.

Here, we prove (9). The inclusion P_X ⊆ F follows from Definition 1. Thus, if (9) does not hold, then there is f₀ ∈ F such that f₀ ∈ P_X,

$\begin{array}{ll}{f}_0(0)=k\text{and }k>0.& (10)\end{array}$

Since X is non-empty, there is (X₀, d₀) ∈ X. Let x₀ be a (unique) point of X₀. Since f₀ belongs to P_X, the function f₀ ∘ d₀ is a metric on X₀. Now, using (10), we obtain

$\begin{array}{l}{f}_0(d_0(x_0,x_0))=f_0(0)=k>0,\end{array}$

which contradicts the positivity property of metric spaces. Inclusion (9) follows.

Let (7) hold. We must show that X is a non-empty subclass of M₁. If X is empty, then

$\begin{array}{ll}{\text{P}}_{\text{X}}=\text{F}& (11)\end{array}$

holds by Theorem 21. Equality (11) contradicts equality (7). Hence, X is non-empty. To complete the proof, we must show that

$\begin{array}{ll}\text{X}\subseteq {\text{M}}_1.& (12)\end{array}$

Let us consider the constant function f₀ : [0, ∞) → [0, ∞) such that

$\begin{array}{ll}{f}_0(t)=0,& (13)\end{array}$

for every t ∈ [0, ∞). Then f₀ belongs to F₀. Hence, for every (X, d) ∈ X, the mapping d₀ : = f₀ ∘ d is a metric on X. Now (13) implies d₀(x, y) = 0 for all x, y ∈ X and (X, d) ∈ X. Hence, card(X) = 1 holds, because the metric space (X, d₀) is one-point by the positivity property. Inclusion (12) follows. The proof is completed.

The next theorem gives us all solutions to the equation P_X = Am.

Theorem 24. The following statements are equivalent for every X ⊆ M.

(i) The inclusion

$\begin{array}{ll}\text{X}\subseteq \text{Dis}& (14)\end{array}$

holds, and there is (Y, ρ) ∈ X with

$\begin{array}{ll}card(Y)⩾2,& (15)\end{array}$

and we have

$\begin{array}{ll}{\text{Dis}}_{X_1}\subseteq \text{X}& (16)\end{array}$

for every (X₁, d₁) ∈ X.

(ii) The equality

$\begin{array}{ll}{\text{P}}_{\text{X}}=\text{A}\text{m}& (17)\end{array}$

holds.

Proof. (i) ⇒ (ii). Let (i) hold. Equality (17) holds if

$\begin{array}{ll}{\text{P}}_{\text{X}}\supseteq \text{Dis}& (18)\end{array}$

and

$\begin{array}{ll}{\text{P}}_{\text{X}}\subseteq \text{Dis}.& (19)\end{array}$

Here, we prove (18). Inclusion (18) holds if we have

$\begin{array}{ll}(X_1,f\circ d_1)\in \text{X}& (20)\end{array}$

for all f ∈ Am and (X₁, d₁) ∈ X. Relation (20) follows from Theorem 23 if (X₁, d₁) ∈ M₁. To see it we only note that Am ⊆ F₀. Let us consider the case when

$card(X_1)⩾2.$

Since (X₁, d₁) is discrete by (14), Definition 6 implies that there is k₁ ∈ (0, ∞) satisfying

$d_1(x,y)=k_1$

for all distinct x, y ∈ X₁. Let f ∈ Am be arbitrary. Then f(k₁) is strictly positive and

$f(d_1(x,y))=f(k_1)$

holds for all distinct x, y ∈ X₁. Thus, f ∘ d₁ is a discrete metric on X₁, i.e., we have

$\begin{array}{ll}(X_1,f\circ d_1)\in {\text{Dis}}_{X_1}.& (21)\end{array}$

Now, Equation (20) follows from Equations (16, 21).

Here, we prove (19). To prove, we must show that every f ∈ P_X is amenable.

Suppose contrary that f belongs to P_X but the equality

$\begin{array}{ll}f(t_1)=0& (22)\end{array}$

holds with some t₁ ∈ (0, ∞). By statement (i) we can find (Y, ρ) ∈ X such that (15) and

$\rho (x,y)=t_1$

hold for all distinct x, y ∈ Y. Now f ∈ P_X and (Y, ρ) ∈ X imply that f ∘ ρ is a metric on Y. Consequently, for all distinct x, y ∈ Y, we have

$f(\rho (x,y))=f(t_1)>0,$

which contradicts (22). The validity of (19) follows.

(ii) ⇒ (i). Let X satisfy equality (17). Since Am ≠ F holds, the class X is non-empty by Theorem 21. Moreover, using Theorem 23, we see that X contains a metric space (X, d) with card(X) ⩾ 2, because Am ≠ F₀.

If the inequality

$card(Y)⩽2$

holds for every (Y, ρ) ∈ X, then all metric spaces belonging to X are discrete (see Example 8). Using the definitions of Dis and Am, it is easy to prove that for each (X₁, d₁) ∈ Dis and every (X₁, d) ∈ Dis_X1 there exists f ∈ Am such that d = f ∘ d₁. Hence to complete the proof, it is suffice to show that every (X, d) ∈ X is discrete when

$\begin{array}{ll}card(X)⩾3.& (23)\end{array}$

Let us consider arbitrary (X, d) ∈ X satisfying (23). Suppose that (X, d) ∉ Dis. Then by Proposition 9 there are distinct a, b, c, ∈ X such that

$\begin{array}{ll}d(b,c)\notin \left\{d(a,b),d(c,a)\right\}.& (24)\end{array}$

Let c₁ and c₂ be points of (0, ∞) such that

$\begin{array}{ll}{c}_2>2c_1.& (25)\end{array}$

Now we can define f ∈ Am as

$\begin{array}{ll}f(t):=\{\begin{array}{ll}0& \text{ if }t=0,\\ c_2& \text{ if }t=d(b,c),\\ c_1& \text{ otherwise}.\end{array}& (26)\end{array}$

Equality (17) implies that f ∘ d is a metric on X. Consequently, we have

$\begin{array}{ll}f(d(b,c))⩽f(d(b,a))+f(d(b,c))& (27)\end{array}$

by triangle inequality. Now using Equations (24, 26) we can rewrite Equation (27) as

$c_2⩽c_1+c_1,$

which contradicts Equation (25). It implies (X, d) ∈ Dis. The proof is completed.

Corollary 25. The equalities

${\text{P}}_{\text{Dis}}={\text{P}}_{{\text{M}}_2}=\text{A}\text{m}$

hold.

Remark 26. The equality

$\begin{array}{l}{\text{P}}_{{\text{M}}_2}=\text{A}\text{m}\end{array}$

is known, see, for example, Remark 1.2 in the article [13]. This article also contains “constructive” characterizations of the smallest bilateral ideal and the largest subgroup of the monoid P_M.

Proposition 27. Let X be a subclass of M. Then P_X is a submonoid of (F, ∘).

Proof. It follows directly from Definition 1 that

${\text{P}}_{\text{X}}\subseteq \text{F}$

holds and that the identity mapping id:[0, ∞) → [0, ∞) belongs to P_X. Hence, by Lemma 19, it is suffice to prove

$\begin{array}{ll}f\circ g\in {\text{P}}_{\text{X}}& (28)\end{array}$

for all f, g ∈ P_X.

Let us consider arbitrary f ∈ P_X and g ∈ P_X. Then, using Definition 1, we see that (X, g ∘ d) belongs to X for every (X, d) ∈ X. Consequently,

$\begin{array}{ll}(X,f\circ (g\circ d))\in \text{X}& (29)\end{array}$

holds. Since the composition of functions is always associative, the equality

$\begin{array}{ll}(f\circ g)\circ d=f\circ (g\circ d)& (30)\end{array}$

holds for every (X, d) ∈ X. Now Equation (28) follows from Equations (29, 30).

The above proposition implies the following corollary.

Corollary 28. If the equation

${\text{P}}_{\text{X}}=\text{A}$

has a solution, then A is a submonoid of F.

The following example shows that the converse of Corollary 28 is, generally speaking, false.

Example 29. Let us define A₁ ⊆ F as

$\begin{array}{l}{\text{A}}_1=\left\{f_1,id\right\},\end{array}$

where f₁ ∈ F is defined such that

$\begin{array}{ll}{f}_1(t):=\{\begin{array}{ll}1& \text{if }t=0,\\ 0& \text{if }t=1,\\ t& \text{otherwise}\end{array}& (31)\end{array}$

and id is the identical mapping of [0, ∞). The equalities f₁ ∘ f₁ = id, f₁ ∘ id = f₁ = id ∘ f₁ show that A₁ is a submonoid of (F, ∘). Suppose that there is X₁ ⊆ M satisfying the equality

$\begin{array}{ll}{\text{P}}_{{\text{X}}_1}={\text{A}}_1.& (32)\end{array}$

Then using Theorem 21, we see that X₁ is non-empty because A₁ ≠ F holds. Let (X₁, d₁) be an arbitrary metric space from A₁. Since X₁ is non-empty, we can find x₁ ∈ X₁. Then (32) implies that f₁ ∘ d₁ is metric on X₁. Now using (31), we obtain

$\begin{array}{l}{f}_1(d_1(x_1,x_1))=f_1(0)=1,\end{array}$

which contradicts the positivity property of metrics.

4 Submonoids of monoids P_M and P_U

The following theorem provides a solution to Problem 2.

Theorem 30. Let A be a non-empty subset of the set P_M of all metric preserving functions. Then the following statements are equivalent.

(i) The equation

$\begin{array}{ll}{\text{P}}_{\text{X}}=\text{A}& (33)\end{array}$

has a solution X ⊆ M.

(ii) A is a submonoid of (F, ∘).

(iii) A is a submonoid of (P_M, ∘).

Proof. (i) ⇒ (ii). Suppose that there is X ⊆ M such that (33) holds.

Then A is a submonoid of (F, ∘) by Proposition 27.

(ii) ⇒ (iii). Let A be a submonoid of (F, ∘). By Proposition 27, the monoid (P_M, ∘) also is a submonoid of (F, ∘). Then using the inclusion A ⊆ P_M, we obtain that A is a submonoid of (P_M, ∘) by Lemma 19.

(iii) ⇒ (i). Let A be a submonoid of (P_M, ∘). We must prove that (33) has a solution X ⊆ M.

Let (X, d) be a metric space such that

$\begin{array}{ll}\left\{d(x,y):x,y\in X\right\}=[0,\infty ).& (34)\end{array}$

$\begin{array}{ll}\text{X}:=\left\{(X,f\circ d):f\in \text{A}\right\}.& (35)\end{array}$

Thus, a metric space (Y, ρ) belongs to X if and only if Y = X and there is f ∈ A such that ρ = f ∘ d.

We claim that Equation (33) holds if X is defined by Equality (35). To prove it, we note that Equation (33) holds if

$\begin{array}{ll}\text{A}\subseteq {\text{P}}_{\text{X}}& (36)\end{array}$

and

$\begin{array}{ll}\text{A}\supseteq {\text{P}}_{\text{X}}.& (37)\end{array}$

Here, we prove Inclusion (36). This inclusion holds if for every f ∈ A and each (Y, ρ) ∈ X we have (Y, f ∘ ρ) ∈ X. Let us consider arbitrary (Y, ρ) ∈ X and f ∈ A. Then, using Equation (35), we can find g ∈ A such that

$\begin{array}{ll}X=Y\text{and }\rho =g\circ d.& (38)\end{array}$

Since A is a monoid, the membership relations f ∈ A and g ∈ A imply g ∘ f ∈ A. Hence, we have

$\begin{array}{ll}(X,g\circ f\circ d)\in \text{X}& (39)\end{array}$

by Equality (35). Now (Y, f ∘ ρ) ∈ X follows from Equations (38, 39).

Here, we prove Inclusion (37). Let g₁ belongs to P_X and let (X, d) be the same as in (35). Then (X, g₁ ∘ d) belongs to X and, using (35), we can find f₁ ∈ A such that

$\begin{array}{ll}(X,g_1\circ d)=(X,f_1\circ d).& (40)\end{array}$

Equality (40) implies

$\begin{array}{ll}{g}_1(d(x,y))=f_1(d(x,y)),& (41)\end{array}$

for all x, y ∈ X. Consequently, g₁(t) = f₁(t) holds for every t ∈ [0, ∞) by Equation (34, 41). Thus, we have g₁ = f₁. That implies g₁ ∈ A. Inclusion (37) follows. The proof is completed.

Remark 31. A reviewer of the article noted that condition (34) can be neatly expressed in terms of center distances which stems from article [40].

Let us turn now to Question 4. Proposition 15 and Lemma 20 provide the following result.

Theorem 32. There is X ⊆ M such that

$\begin{array}{ll}{\text{P}}_{\text{X}}=\text{S}\text{I}.& (42)\end{array}$

Proof. By Proposition 27, the monoids (P_M, ∘) and (P_U, ∘) are submonoids of (F, ∘). The equality

$\begin{array}{ll}\text{S}\text{I}={\text{P}}_{\text{M}}\cap {\text{P}}_{\text{U}}& (43)\end{array}$

holds by Proposition 15. Using Equality (43) and Lemma 20 with T₁ = P_M, T₂ = P_U, and S = F, we see that SI also is a submonoid of F. Consequently, Theorem 30 with A = SI implies that there is X ⊆ M such that (42) holds.

The next theorem is an ultrametric analog of Theorem 30 and it gives us a solution to Problem 3.

Theorem 33. Let A be a non-empty subset of the set P_U of all ultrametric preserving functions. Then the following statements are equivalent.

(i) The equation P_X = A has a solution X ⊆ U.

(ii) A is a submonoid of (F, ∘).

(iii) A is a submonoid of (P_U, ∘).

A proof of Theorem 33 can be obtained by a simple modification of the proof of Theorem 30. We only note that the ultrametric space defined in Example 5 satisfies equality (34) with $X={ℝ}_0^{+}$ and d = d⁺.

5 Two conjectures

Conjecture 34. The equation

$\begin{array}{l}{\text{P}}_{\text{X}}=\text{A}\end{array}$

has a solution X ⊆ M for every submonoid A of the monoid Am.

Example 29 shows that we cannot replace Am with F in Conjecture 34, but we hope that the following is valid.

Conjecture 35. For every submonoid A of the monoid F, there exists X ⊆ M such that P_X and A are isomorphic submonoids.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

VB: Writing – original draft, Writing – review & editing. OD: Methodology, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing.

Funding

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. VB was partially supported by a grant from the Simons Foundation (Award 1160640, Presidential Discretionary-Ukraine Support Grants, VB). OD was supported by grant 359772 of the Academy of Finland.

Acknowledgments

We express our deep gratitude toward the reviewers for the careful study of our paper. Their elaborate, constructive criticism allowed us to improve this paper and suggested further directions of the research.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

1. Jachymski J, Turoboś F. On functions preserving regular semimetrics and quasimetrics satisfying the relaxed polygonal inequality. Rev R Acad Cienc Exactas Fís Nat, Ser A Mat. (2020) 114:159. doi: 10.1007/s13398-020-00891-7

Crossref Full Text | Google Scholar

2. Dovgoshey O. Strongly ultrametric preserving functions. Topol Appl. (2024) 351:108931. doi: 10.1016/j.topol.2024.108931

PubMed Abstract | Crossref Full Text | Google Scholar

3. Delhommé C, Laflamme C, Pouzet M, Sauer N. Indivisible ultrametric spaces. Topol Appl. (2008) 155:1462–78. doi: 10.1016/j.topol.2008.03.012

Crossref Full Text | Google Scholar

4. Searcóid MO. Metric Spaces. Berlin: Springer (2006).

Google Scholar

5. Wilson WA. On certain types of continuous transformations of metric spaces. Am J Mathem. (1935) 57:62–8. doi: 10.2307/2372019

Crossref Full Text | Google Scholar

6. Blumenthal LM. Remarks concerning the Euclidean four-point property. Ergeb Math Kolloq Wien. (1936) 7:7–10.

Google Scholar

7. Borsík J, Doboš J. Functions the composition with a metric of which is a metric. Math Slovaca. (1981) 31:3–12.

Google Scholar

8. Borsík J, Doboš J. On metric preserving functions. Real Anal Exch. (1988) 13:285–93. doi: 10.2307/44151879

Crossref Full Text | Google Scholar

9. Doboš J. On modification of the Euclidean metric on reals. Tatra Mt Math Publ. (1996) 8:51–4.

Google Scholar

10. Doboš J, Piotrowski Z. Some remarks on metric-preserving functions. Real Anal Exch. (1994). 19:317–20. doi: 10.2307/44153846

Crossref Full Text | Google Scholar

11. Doboš J, Piotrowski Z. A note on metric preserving functions. Int J Math Math Sci. (1996) 19:199–200. doi: 10.1155/S0161171296000282

Crossref Full Text | Google Scholar

12. Doboš J, Piotrowski Z. When distance means money. Internat J Math Ed Sci Tech. (1997) 28:513–8. doi: 10.1080/0020739970280405

Crossref Full Text | Google Scholar

13. Dovgoshey O, Martio O. Functions transferring metrics to metrics. Beitráge zur Algebra und Geometrie. (2013) 54:237–61. doi: 10.1007/s13366-011-0061-7

Crossref Full Text | Google Scholar

14. Vallin RW. Metric preserving functions and differentiation. Real Anal Exch. (1997) 22:86. doi: 10.2307/44152729

Crossref Full Text | Google Scholar

15. Vallin RW. On metric preserving functions and infinite derivatives. Acta Math Univ Comen, New Ser. (1998) 67:373–6.

PubMed Abstract | Google Scholar

16. Vallin RW. A subset of metric preserving functions. Int J Math Math Sci. (1998) 21:409–10. doi: 10.1155/S0161171298000568

Crossref Full Text | Google Scholar

17. Vallin RW. Continuity and differentiability aspects of metric preserving functions. Real Anal Exch. (1999/2000) 25:849–68. doi: 10.2307/44154040

Crossref Full Text | Google Scholar

18. Pongsriiam P, Termwuttipong I. On metric-preserving functions and fixed point theorems. Fixed Point Theory Appl. (2014) 2014:14. doi: 10.1186/1687-1812-2014-179

Crossref Full Text | Google Scholar

19. Vallin RW. On preserving (ℝ, Eucl.) and almost periodic functions. Tatra Mt Math Publ. (2002) 24:1–6.

Google Scholar

20. Doboš J. Metric Preserving Functions. Košice, Slovakia: Štroffek (1998).

Google Scholar

21. Corazza P. Introduction to metric-preserving functions. Amer Math Monthly. (1999) 104:309–23. doi: 10.1080/00029890.1999.12005048

Crossref Full Text | Google Scholar

22. Pongsriiam P, Termwuttipong I. Remarks on ultrametrics and metric-preserving functions. Abstr Appl Anal. (2014) 2014:1–9. doi: 10.1155/2014/163258

Crossref Full Text | Google Scholar

23. Dovgoshey O. On ultrametric-preserving functions. Math Slovaca. (2020) 70:173–82. doi: 10.1515/ms-2017-0342

Crossref Full Text | Google Scholar

24. Vallin RW, Dovgoshey OA. P-adic metric preserving functions and their analogues. Math Slovaca. (2021) 71:391–408. doi: 10.1515/ms-2017-0476

Crossref Full Text | Google Scholar

25. Dovgoshey O. Combinatorial properties of ultrametrics and generalized ultrametrics. Bull Belg Math Soc Simon Stevin. (2020) 27:379–417. doi: 10.36045/bbms/1599616821

Crossref Full Text | Google Scholar

26. Priess-Crampe S, Ribenboim P. Fixed points, combs and generalized power series. Abh Math Sem Univ Hamburg. (1993) 63:227–44. doi: 10.1007/BF02941344

Crossref Full Text | Google Scholar

27. Priess-Crampe S, Ribenboim P. Generalized ultrametric spaces I. Abh Math Sem Univ Hamburg. (1996) 66:55–73. doi: 10.1007/BF02940794

Crossref Full Text | Google Scholar

28. Priess-Crampe S, Ribenboim P. Generalized ultrametric spaces II. Abh Math Sem Univ Hamburg. (1997) 67:19–31. doi: 10.1007/BF02940817

Crossref Full Text | Google Scholar

29. Ribenboim P. The new theory of ultrametric spaces. Periodica Math Hung. (1996) 32:103–11. doi: 10.1007/BF01879736

Crossref Full Text | Google Scholar

30. Ribenboim P. The immersion of ultrametric spaces into Hahn Spaces. J Algebra. (2009) 323:1482–93. doi: 10.1016/j.jalgebra.2009.11.032

Crossref Full Text | Google Scholar

31. Borsík J, Doboš J. On a product of metric spaces. Math Slovaca. (1981) 31:193–205.

Google Scholar

32. Bernig A, Foertsch T, Schroeder V. Non standard metric products. Beitráge zur Algebra und Geometrie. (2003) 44:499–510.

Google Scholar

33. Dovgoshey O, Petrov E, Kozub G. Metric products and continuation of isotone functions. Math Slovaca. (2014) 64:187–208. doi: 10.2478/s12175-013-0195-1

Crossref Full Text | Google Scholar

34. Foertsch T, Schroeder V. Minkowski-versus Euclidean rank for products of metric spaces. Adv Geom. (2002) 2:123–31. doi: 10.1515/advg.2002.002

Crossref Full Text | Google Scholar

35. Herburt I. Moszyńska M. On metric products. Colloq Math. (1991) 62:121–33. doi: 10.4064/cm-62-1-121-133

PubMed Abstract | Crossref Full Text | Google Scholar

36. Kazukawa D. Construction of product spaces. Anal Geom Metr Spaces. (2021) 9:186–218. doi: 10.1515/agms-2020-0129

Crossref Full Text | Google Scholar

37. Lichman M, Nowakowski P, Turobo F. On Functions Preserving Products of Certain Classes of Semimetric Spaces. Tatra Mount Mathem Publicat. (2021) 78:175–98. doi: 10.2478/tmmp-2021-0013

Crossref Full Text | Google Scholar

38. Dovgoshey O, Martio O. Products of metric spaces, covering numbers, packing numbers and characterizations of ultrametric spaces. Rev Roumaine Math Pures Appl. (2009) 54:423–39.

Google Scholar

39. Howie JM. Fundamentals of Semigroup Theory. Oxford: Clarendan Press. (1995).

Google Scholar

40. Bielas W, Plewik S, Walczyska M. On the center of distances. Eur J Mathem. (2018) 4:687–98. doi: 10.1007/s40879-017-0199-4

Crossref Full Text | Google Scholar