Pseudometric spaces. From minimality to maximality in the groups of combinatorial self-similarities

The group of combinatorial self-similarities of a pseudometric space $(X, d)$ is the maximal subgroup of the symmetric group $\mathbf{Sym} (X)$ whose elements preserve the four-point equality $d(x,y)=d(u,v)$. Let us denote by $\mathcal{IP}$ the class of all pseudometric spaces $(X, d)$ for which every combinatorial self-similarity $\Phi\colon~X~\to~X$ satisfies the equality $d(x, \Phi(x))=0,$ but all permutations of metric reflection of $(X, d)$ are combinatorial self-similarities of this reflection. The structure of $\mathcal{IP}$ spaces is fully described.

The concept of combinatorial similarity for pseudometrics and, more generally, for mappings deffined on the Cartesian square of a set, was introduced in [3].Theorem 3.1 [3] contains a necessary and sufficient condition under which a mapping is combinatorially similar to a pseudometric.The strongly rigid pseudometrics and discrete ones, were characterized, up to combinatorial similarity, in Theorems 4.13 and 3.9 of [3], respectively.
The main result of [5], Theorem 2.8, completely, describes the structure of semimetric spaces (X, d) for which the group of combinatorial self-similarities of (X, d) coincides with the symmetric group Sym(X).In particular Theorem 2.8 shows that every permutation of X is a combinatorial self-similarity of (X, d) if d is discrete or strongly rigid.
The above mentioned results are the starting points of our research.To describe the general direction of this research we note that the transition from the pseudometric space (X, d) to its metric reflection (X/ 0 = , δ d ) naturally generates a homomorphism H : Cs(X, d) → Cs(X/ 0 = , δ d ) of the groups of combinatorial self-similarities of (X, d) and (X/ 0 = , δ d ).How are the algebraic properties of the homomorphism H related to the metric properties of (X, d)?
In the present paper we give a metric characterization of pseudometric spaces (X, d) for which the kernel of H coincides with Cs(X, d) but the equality Cs(X/ 0 = , δ d ) = Sym(X/ 0 = ) holds.

Preliminaries
Let us start from the classical notion of metric space.A metric on a set X is a function d : X 2 → R such that for all x, y, z ∈ X: (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.In 1934 Duro Kurepa [6] introduced the pseudometric spaces which, unlike metric spaces, allow the zero distance between different points.Definition 2.1.Let X be a set and let d : X 2 → R be a non-negative, symmetric function such that d(x, x) = 0 for every x ∈ X.The function d is a pseudometric on X if it satisfies the triangle inequality.
If d is a pseudometric on X, we say that (X, d) is a pseudometric space.We will denote by d(X 2 ) the range of the pseudometric d, d(X 2 ) := {d(x, y) : x, y ∈ X}.
Thus the notion of combinatorial similarities can be considered as a generalization of the notion of the isometries of metric spaces.
Remark 2.4.The notion of isometry of metric spaces can be extended to pseudometric spaces in various non-equivalent ways.For example, John Kelley [7] define the isometries of pseudometric spaces (X, d) and (Y, ρ) as the distance-preserving surjections X → Y .
We say that two pseudometric spaces are pseudoisometric if there is a pseudoisometry of these spaces.
For every pseudometric spaces (X, d) the set of all combinatorial selfsimilarities is a group with the function composition as a group operation.The identity mapping, Id X : X → X, Id X (x) = x for every x ∈ X, is the identity element of this group.If d : X 2 → R is a metric, then we can characterize Id X as a unique mapping X f − → X which satisfies the equality (2) d(x, f (x)) = 0 for every x ∈ X.For the case when the pseudometric d : X 2 → R is not a metric, one can always find a bijection X f − → X such that f = Id X but (2) holds for every x ∈ X.
Example 2.6.Let (X, d 0 ) be a pseudometric space endowed by the zero pseudometric, d(x, y) = 0 for all x, y ∈ X.Then (2) holds for every x ∈ X and each Remark 2.8.It is clear that, for every pseudometric space (X, d), the set of all pseudoidentities X → X is a subgroup of the group of all combinatorial self-similarities of (X, d).
The combinatorial similarities of pseudometric spaces are the main type of morphisms studied in this paper.
The groups of all combinatorial self-similarities and all pseudoidentities of a pseudometric space (X, d) will be denoted by Cs(X, d) and PI(X, d) respectively.Thus, for every pseudometric space (X, d) we have where Sym(X) is the symmetric group of all permutations of the set X.
For every nonempty pseudometric space (X, d), we define a binary relation 0(d)  = on X by for all x, y ∈ X.
In the future, we will simply write 0 = instead of 0(d) = , when it is clear which d we are talking about.
The proof of the following proposition can be found in [7, Chapter 4, Theorem 15].Proposition 2.9.Let X be a nonempty set and let d : X 2 → R be a pseudometric on X.Then 0 = is an equivalence relation on X and, in addition, the function δ d , (4) is a correctly defined metric on the quotient set X/ 0 = .
In what follows we will say that the metric space (X/ 0 = , δ d ) is the metric reflection of (X, d).  6) Thus, (X, d) ∈ IP holds if and only if the group Cs(X, d) is as small as possible, but the group Cs(X/ 0 = , δ d ) is as large as possible.
Example 2.12.Let (X, d) be a nonempty metric space.Then (X, d) ∈ IP holds if and only if |X| = 1.Indeed, the implication is evidently valid.Let (X, d) belong to IP.To prove the equality |X| = 1 we note that PI(X, d) contains the mapping Id X : X → X only and that Cs(X, d) and Cs(X/ 0 = , δ d ) are isomorphic groups because (X, d) is a metric space.Hence, (5)  The main goal of the paper is to describe the structure of pseudometric spaces belonging to IP.To do this, we introduce into consideration pseudometric generalizations of some well-known classes of metric spaces.
Let (X, d) be a metric space.Recall that the metric d is said to be strongly rigid if, for all x, y, u, v ∈ X, the condition (Some properties of strongly rigid metric spaces are described in [3,[8][9][10][11][12][13][14][15][16].) The concept of strongly rigid metric can be naturally generalized to the concept of strongly rigid pseudometric what was done in paper [3].Definition 2.13.Let (X, d) be a pseudometric space.The pseudometric d is strongly rigid if every metric subspace of (X, d) is strongly rigid.
Remark 2.14.A pseudometric d : X 2 → R is strongly rigid if and only if (7) implies Example 2.15.The implication (7) ⇒ (9) is vacuously true for the zero pseudometric d : X 2 → R. Hence, the zero pseudometric is strongly rigid.The well-known example of pseudometric space is a seminormed vector space.Recall that a semimorm on a vector space X is a function • : X → R such that x + y x + y and αx = |α| x for all x, y ∈ X and every scalar α.If (X, • ) is a seminormed vector space then the function ρ : X 2 → R, ρ(x, y) = x − y , is a pseudometric on X.
In the following example we construct a strongly rigid pseudometric subspace of a seminormed real vector space.
Example 2.16.Let (R 2 , • ) be a two-demensional seminormed real vector space endowed with the seminorm • such that x, y = |x| for each x, y ∈ R 2 .The field R of real numbers can be also considered as a vector space over the field Q of rational numbers.Let H be a maximal linearly independent over Q subset of R. If we define a subset X of R 2 as X = { x, y ∈ R 2 : x ∈ H and y ∈ R}, then X is a strongly rigid pseudometric subspace of the seminormed space (R 2 , • ).
We say that a metric d : holds, where |d(X 2 )| is the cardinal number of the set d(X 2 ).
Remark 2.17.The standard definition of discrete metric can be formulated as: "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, [17, p. 4].) The following definition is a suitable reformulation of the corresponding concept from [3].Definition 2.18.Let (X, d) be a pseudometric space.The pseudometric d is discrete if all metric subspaces of (X, d) are discrete.
Remark 2.19.If is easy to see that pseudometric d : Definition 2.20.A pseudometric space (X, d) is a pseudorectangle if all three-point metric subspaces of (X, d) are strongly rigid and isometric and, in addition, there is a four-point metric subspace Y of (X, d) such that for every x ∈ X we can find y ∈ Y satisfying d(x, y) = 0.
It is easy to see that the metric reflection (X/ 0 = , δ d ) of every pseudorectangle (X, d) is a four-point metric space and, in addition, it can be shown that this metric reflection is combinatorially similar to the vertex set of Euclidean non-square rectangle.
The paper is organized as follows.
In the next section we recall some known interconnections between equivalence relations, partitions of sets and discrete pseudometrics.A simple sufficient condition for the equality Cs(X, d) = PI(X, d) is found in Corollary 3.6 of Proposition 3.4.
The main results of the paper are given in Sections 3 and 4. A complete description of pseudometric spaces (X, d) that satisfy the equality Cs(X/ 0 = , δ d ) = Sym(X/ 0 = ) is given in Theorem 4.4.This theorem together with Corollary 3.6 allow us to characterize IP-spaces in Theorem 4.6.
A combinatorial characterization of fibers of pseudometrics is proved in Theorems 5.3, 5.4 and 5.6 for strongly rigid spaces, discrete spaces and pseudorectangles, respectively.As a corollary of these theorems we obtain a new description of IP-spaces in Theorem 5.7.In Proposition 5.9 we show that pseudorectangles or strongly rigid spaces (X, d) and (Y, ρ) are combinatorially similar if and only if the binary relations 0(d)  = and = are the same.The characteristic properties of 0 = are described in Proposition 5.10 for pseudorectangles and strongly rigid pseudometric spaces.
The final result of the paper, Theorem 5.12, characterizes the class of discrete pseudometric spaces, strongly rigid pseudometric spaces and pseudorectangles in terms of same extremal properties of these classes.

Partitions of sets
Let U be a set.A binary relation on U is a subset of the Cartesian square U 2 = U × U = { x, y : x, y ∈ U}.
A binary relation R ⊆ U 2 is an equivalence relation on U if the following conditions hold for all x, y, z ∈ U: (i) x, x ∈ R, the reflexivity property; (ii) ( x, y ∈ R) ⇔ ( y, x ∈ R), the symmetry property; (iii) (( x, y ∈ R) and ( y, z ∈ R)) ⇒ ( x, z ∈ R), the transitivity property.If R is an equivalence relation on U, then an equivalence class is a subset of U having the form (10) [a] R = {x ∈ U : x, a ∈ R} for some a ∈ U.The quotient set U/R of U with respect to R is the set of all equivalence classes [a] R .Let X be a nonempty set and P = {X j : j ∈ J} be a set of nonempty subsets of X.The set P is a partition of X with the blocks X j , j ∈ J, if ∪ j∈J X j = X and X j 1 ∩ X j 2 = ∅ for all distinct j 1 , j 2 ∈ J. Definition 3.1.Partitions P and Q of a set X are equal, P = Q, if every block of P is a block of Q and vice versa.
Every partition P of a set X is a subset of the power set 2 X , P ⊆ 2 X , and each block of P is a point of 2 X .Thus, Definition 3.1 simply means that P = Q holds if and only if P and Q are the same subsets of 2 X .Consequently, P = Q holds if and only if P ⊆ Q and Q ⊆ P .The following lemma states that any of the above inclusions is sufficient for 13) and the definition of partitions of sets we obtain the contradiction,

Equality (11) follows.
There exists the well-known, one-to-one correspondence between the equivalence relations on sets and the partitions of sets (see, for example, [18,Chapter II,§ 5] or [19,Theorem 1]).Proposition 3.3.Let X be a nonempty set.If P = {X j : j ∈ J} is a partition of X and R is a binary relation on X such that the logical equivalence x, y ∈ R ⇔ ∃j ∈ J : x ∈ X j and y ∈ X j is valid for every x, y ∈ X 2 , then R is an equivalence relation on X with the quotient set P and it is the unique equivalence relation on X having P as the quotient set.Conversely, if R is an equivalence relation on X, then there is the unique partition P of X such that P is the quotient set of X with respect to R.
The next proposition shows, in particular, that if Ψ : Y → X is a combinatorial similarity of pseudometric spaces (X, d) and (Y, ρ), then the equivalence classes of the relation 0(d)  = are the images of the equivalence classes of 0(ρ)  = under mapping Ψ.
Proposition 3.4.Let (X, d) and (Y, ρ) be nonempty pseudometric spaces and let Ψ : Y → X be a combinatorial similarity of these spaces.If for all x, y ∈ Y , then the equalities Proof.Using Definition 2.1 and equality ( 14) with y = x we obtain that implies (15).
Corollary 3.5.Let d and ρ be two combinatorially similar pseudometrics defined on the same nonempty set.Then the binary relations 0(d) = and 0(ρ) = are equal.
The next corollary gives a simple sufficient condition under which equality (5) holds.
Corollary 3.6.Let (X, d) be a nonempty pseudometric space and let {X j : j ∈ J} be a partition of Xcorresponding the equivalence relation holds for all different j 1 , j 2 ∈ J. Let us consider an arbitrary combinatorial self-similarity Ψ : X → X of (X, d).We must show that Ψ is a pseudoidentity of (X, d).To do it, we rewrite equality ( 16) in the form Since Ψ is a bijective mapping, |Ψ(X j )| = |X j | holds for every j ∈ J. Now, Ψ ∈ PI(X, d) follows from ( 20) and ( 21).
The next result directly follows from Proposition 3.6 and Corollary 3.7 of [3], and shows that a partial converse to Proposition 2.9 is also true.Proposition 3.7.Let X be a nonempty set and let ≡ be an equivalence relation on X.Then there is a unique up to combinatorial similarity discrete pseudometric d : is valid for all x, y ∈ X. = and 0(ρ) = are the same.
In the next section of the paper we will prove that the equality of binary relations 0(d)  = and 0(ρ) = is equivalent to combinatorial similarity of (X, d) and (X, ρ), when both spaces are pseudorectangles or strongly rigid pseudometric spaces.

Structure of IP spaces
The following theorem is a special case of Theorem 2.8 [5], which gives us a complete description of semimetric spaces satisfying the equality Cs(X, d) = Sym(X).Theorem 4.1.Let (X, d) be a nonempty metric space.Then the following statements are equivalent: (i) At least one of the following conditions has been fulfilled: All three-point subspaces of (X, d) are strongly rigid and isometric.(ii) The equality Cs(X, d) = Sym(X) holds.
Our initial objective is to find a "pseudometric" analog of this theorem.
The next lemma shows that the transition from pseudometric space to its metric reflection preserves the concept of discreteness, strong rigidity, and the property of being a pseudorectangle.Lemma 4.2.Let (X, d) be a nonempty pseudometric space.Then the following statements hold: = , δ d ) contains exactly four points and all three-point subspaces of (X/ 0 = , δ d ) are strongly rigid and isometric.
Proof.Let (X, d) be strongly rigid.Then, using the Axiom of Choice (AC), we find a metric subspace Y of the pseudometric space (X, d) such that for every x ∈ X there is the unique y ∈ Y which satisfies d(x, y) = 0. Since (X, d) is strongly rigid, the metric space Y is also strongly rigid by Definition 2.13.Moreover, Proposition 2.9 implies that the metric spaces Y and (X/ 0 = , δ d ) are isometric.Hence, (X/ 0 = , δ d ) is strongly rigid.To complete the proof of statement (i) we must show that the strong rigidity of (X/ 0 = , δ d ) implies that (X, d) is also a strongly rigid.Let (X/ 0 = , δ d ) be strongly rigid and let Z be a metric subspace of (X, d).Using AC, we find a metric subspace Y Z of (X, d) such that Y Z ⊇ Z and for every x ∈ X there is the unique y ∈ Y Z , which satisfies d(x, y) = 0. Proceeding as above, we can see that Y Z and (X/ 0 = , δ d ) are isometric.Hence, Y Z is strongly rigid and, consequently, being a subspace of Y X , Z is also strongly rigid.
Let us prove statement (ii).Let π : X → X/ 0 = be the canonical projection, π(x) : = {y ∈ X : d(x, y) = 0}.Then, by formula (4) of Proposition 2.9, the equality holds for all x, y ∈ X.Using (23), we see that d and δ d has one and the same range, and, consequently (ii) holds by Remark 2.19.The validity of statement (iii) follows from Proposition 2.9 and Definition 2.13.
The following lemma is, in fact, a particular case of Proposition 2.3 from [5].(i) All three-point subspaces of (X, d) are strongly rigid and isometric.(ii) (X, d) is combinatorially similar to the space of vertices of Euclidean non-square rectangle.
Now we are ready to characterize the pseudometric spaces satisfying equality (6).
Theorem 4.4.Let (X, d) be a nonempty pseudometric space.Then the following statements are equivalent: (i) At least one of the following conditions has been fulfilled: Cs(X/ 0 = , δ d ) = Sym(X/ 0 = ) holds.
The next proposition describes the combinatorial self-similarities of a pseudometric space (X, d) via combinatorial self-similarities of the metric reflection (X/ 0 = , δ d ) of this space.
Proposition 4.5.Let (X, d) be a pseudometric space, let Φ : X → X be a bijective mapping and let π : X → X/ 0 = be the canonical projection, If there is a combinatorial self-similarity Ψ : Proof.Let Ψ be a combinatorial self-similarity of (X/ 0 = , δ d ).Then, using Definition 2.2, we find a bijection f : is commutative, where also is a commutative diagram, where for every x, y ∈ X 2 .The commutativity of ( 26) and (27) implies that (28) ) and, in addition, this proposition implies the equality of mappings Hence, the commutativity of (28) gives us the commutativity of the diagram . By Definition 2.2, the last diagram is commutative iff Φ : X → X is a combinatorial self-similarity of (X, d).
The next theorem can be considered as one of the main results of the paper.
Theorem 4.6.Let (X, d) be a nonempty pseudometric space and let {X j : j ∈ J} be a partition of Xcorresponding the equivalence relation holds whenever j 1 , j 2 ∈ J are distinct and, in addition, at least one of the following conditions has been fulfilled: Proof.Suppose that (29) holds whenever j 1 , j 2 ∈ J are distinct, and that al least one from conditions (i) − (iii) has been fulfilled.Then the membership (X, d) ∈ IP follows from Corollary 3.5 and Theorem 4.4.Let (X, d) belong to IP.Then the equality (30) Cs(X/ 0 = , δ d ) = Sym(X/ 0 = ) holds and, consequently, at least one from conditions (i) − (iii) is valid.Thus, to complete the proof it suffices to show that (29) is valid for all distinct j 1 , j 2 ∈ J. Suppose contrary that there exist j 1 , j 2 ∈ J such that Then there is a bijection Φ : X → X which satisfies the equalities (31) Φ(X j 1 ) = X j 2 and Φ(X j ) = X j whenever j ∈ J and j 1 = j = j 2 .Let x j 1 and x j 2 be some points of X j 1 and X j 2 respectively.Write (32) x * 1 = π(x j 1 ) and where π is the canonical projection of X on X/ 0 = and define a bijection is commutative.Moreover, the mapping Ψ is a combinatorial selfsimilarity of (X/ 0 = , δ d ) by (30).Hence, Φ is a combinatorial self-similarity of (X, d) by Proposition 4.5.Definition 2.7 and (31) imply Φ ∈ PI(X, d).
We conclude this section with the following open problem closely related to Theorem 4.4 and Theorem 4.6.

From partitions of X to partitions of X 2
It is well-known that for every nonempty set X and arbitrary surjection f : X → Y the family In what follows we set (34) for every nonempty pseudometric space (X, d).
In this section we describe the structure of the partition P d −1 , when (X, d) is strongly rigid or discrete, or (X, d) is a pseudorectangle.This allows us to obtain new characteristics of IP spaces and expand Corollary 3.8 to strongly rigid pseudometric spaces and pseudorectangles.
The following lemma gives a "constructive variant" of Proposition 3.3.
Lemma 5.1.Let X be a nonempty set and let P = {X j : j ∈ J} be a partition of X.If R is the equivalence relation corresponding to P , then the equality R = ∪ j∈J X 2 j holds.For the proof of Lemma 5.1 see, for example, Theorem 6 in [7, p. 9].Proposition 5.2.Let (X, d) be a pseudometric space and {X j : j ∈ J} be the quotient set of X with respect to the equivalence relation 0 = .Then for any fixed non-zero element t 0 of d(X 2 ), the following statements are equivalent: (i) There are different j 1 , j 2 ∈ J such that (ii) The assertion is valid whenever Proof.(i) ⇒ (ii).Let different j 1 , j 2 ∈ J satisfy (35).We must show that (37) implies (36) for all x, y, u, v ∈ X. Suppose (37) holds.Then we have x, y , u, v ∈ d −1 (t 0 ).
Since the sets X j 1 × X j 2 and X j 2 × X j 1 are disjoint, (35) implies that only the following cases are possible: x, y ∈ X j 1 × X j 2 and u, v ∈ X j 1 × X j 2 , (38) The sets X j 1 and X j 2 are different elements of the quotient set X/ 0 = .Hence, each of (38) and (39) implies x 0 = u and y 0 = v.Analogously, x 0 = v and y 0 = u hold whenever (40) or ( 41) is valid.Thus, (ii) holds.(ii) ⇒ (i).Let (ii) hold and let x, y be points of X satisfying (42) d(x, y) = t 0 .
Comparing Definition 2.13 with statement (ii) of Proposition 5.2, we see that (ii) can be considered as a singular version of the global property "to be strongly rigid".In Theorem 5.3 below we characterize the strong rigidity of pseudometrics by "globalization" of statement (i) of Proposition 5.2.
Let Q = {X j : j ∈ J} be a partition of a nonempty set X. Then we define a partition Q ⊗ 1 Q of X 2 by the rule: The next theorem follows directly from Theorem 4.13 and Corollary 4.14 of [3].
Theorem 5.3.Let (X, d) be a nonempty pseudometric space.Then the following conditions are equivalent: (iii) There is a partition Q of X such that (45) holds.
The following result is an analog of Theorem 5.3 for discrete pseudometrics.
Theorem 5.4.Let (X, d) be a nonempty pseudometric space.Then the following conditions are equivalent: (iii) There is a partition Q of X such that (46) holds.
Proof.The implications (i) ⇒ (ii) ⇒ (iii) ⇒ (i) are evidently valid if d is the zero pseudometric on X.Let us consider the case when |d(X 2 )| 2.
(i) ⇒ (ii).Let d be a discrete pseudometric and let be the partition of X corresponding to the relation 0(d) = .By Lemma 5.1, the equality It was noted in Remark 2.19 that the inequality |d(X 2 )| 2 holds for discrete d.The last inequality and |d(X 2 )| 2 imply that |d(X 2 )| = 2. Thus, the partition P d −1 of X 2 contains exactly two blocks.Since one of this block is d −1 (0), the second one coincides with X 2 \ j∈J X 2 j by (47).Now (ii) follows from the definition of Q ⊗ 2 Q.
(ii) ⇒ (iii).This implication is trivially valid.(iii) ⇒ (i).Let a partition Q = {X j : j ∈ J} of X satisfy equality (46).Then this equality and the definition of Using Remark 2.19, we see that d is a discrete pseudometric.
Remark 5.5.Theorem 5.4 can be considered as a special case of Theorem 3.9 from [3], that describes all mappings with domain X 2 which are combinatorially similar to discrete pseudometrics on X.
Let X be a set with |X| 4 and let Q = {X 1 , X 2 , X 3 , X 4 } be a partition of the set X. Let us denote by Q ⊗ 3 Q a partition of X 2 having the blocks: Theorem 5.6.Let (X, d) be a nonempty pseudometric space.Then the following conditions are equivalent: (iii) There is a partition Q of X such that (51) holds.
Proof.(i) ⇒ (ii).Let (X, d) be a pseudorectangle.By Definition 2.20, there is a set Y = {y 1 , y 2 , y 3 , y 4 }, such that Y ⊆ X and the sets are the equivalence classes of the relation 0 = .We claim that (51) holds if To prove the last claim we first show that P d −1 contains exactly four blocks, i.e.
(54) |d(X 2 )| = 4 holds.Indeed, by Definition 2.20, all three-point metric subspaces of (X, d) are isometric that implies Moreover, it is easy to see that a finite nonempty metric space (Z, ρ) is strongly rigid if and only if the number of two-point subsets of Z is the same as the number of non-zero elements of ρ(Z 2 ), In particular, a three-point metric subspace S of the pseudometric space (X, d) is strongly rigid if and only if |d(S 2 )| = 4. Consequently, Definition 2.20 implies whenever S is a three-point metric subspace of (X, d).Now, equality (54) follows from (55) and (56).
Let us prove (51).By Lemma 3.2, it suffices to show that the inclusion ) and t 1 > 0 hold.Then there exist two different points y i 1 , y i 2 ∈ {y 1 , y 2 , y 3 , y 4 } such that d(y i 1 , y i 2 ) = t 1 .Without loss the generality, we assume that i 1 = 1 and i 2 = 2.Then, by (53), we have Since all triangles of the metric space {y 1 , y 2 , y 3 , y 4 } are isometric and strongly rigid, Lemma 4. Similarly (58), equality (59) implies and, consequently, If the last inclusion is strict, then there are points are satisfied, contrary to (60).Thus, the equality holds, that together with i implies (57) for every t ∈ d(X 2 ).
As in the proof of Theorem 5.3, it can be shown that Moreover, the definitions of Q ⊗ 3 Q and (51) imply the equality Let us consider a four-point set Y = {y 1 , y 2 , y 3 , y 4 } such that y i ∈ X i holds for every i ∈ {1, 2, 3, 4}.Then, using (61), we see that Y is a fourpoint metric subspace of (X, d) and, for every x ∈ X, there is y ∈ Y such that d(x, y) = 0. Now, by Definition 2.20, (X, d) is a pseudorectangle if and only if all three-point metric subspaces of (X, d) are strongly rigid and isometric.Hence, using (62), we see that (X, d) is a pseudorectangle if and only if holds for every three-point metric subspace Z of (X, d).Let us consider arbitrary z 1 , z 2 , z 3 ∈ X such that d(z i , z j ) = 0 for all distinct i, j ∈ {1, 2, 3}.Since every permutation preserves the partition Q ⊗ 3 Q of the set X 2 , we can assume that z 1 ∈ X 1 , z 2 ∈ X 2 and z 3 ∈ X 3 .Now (63) follows from (48), ( 49) and (50).
Using Theorem 5.3 and Theorems 5.4 and 5.6, we obtain the following modification of Theorem 4.6.
Theorem 5.7.Let (X, d) be a nonempty pseudometric space and let Q = {X j : j ∈ J} be a partition of X corresponding to the equivalence relation 0(d)  = .Then (X, d) ∈ IP if and only if Let us extend Corollary 3.8 to pseudorectangles and strongly rigid pseudometric spaces.Lemma 5.8.Let (X, d) and (X, ρ) be nonempty pseudometric spaces.If the equality (64) holds, then the identical mapping Id X : X → X is a combinatorial similarity of (X, d) and (X, ρ).
Proof.Let (64) hold.Then by (34) we have and, consequently, there is a bijection f : Using (65) it is easy to see that the diagram (66) The mapping Id X 2 coincides with the mapping Id X ⊗ Id X , Id X 2 ( x, y ) = x, y = Id X (x), Id X (y) holds for every x, y ∈ X 2 .Thus the commutativity of (66) implies the commutativity of By Definition 2.2, the last diagram is commutative if and only if the mapping Id X : X → X is a combinatorial similarity of (X, d) and (X, ρ).Proposition 5.9.Let (X, d) and (X, ρ) be either pseudorectangles or nonempty strongly rigid pseudometric spaces.Then the following statements are equivalent: (i) The pseudometric spaces (X, d) and (X, ρ) are combinatorially similar.(ii) The binary relations 0(d)  = and 0(ρ) = are the same.
Let Q = {X j : j ∈ J} be the partition of X corresponding the equivalence relation 0(d)  = .Then, by Theorem 5.3, we have (67) Since the relations 0(d) = and 0(ρ) = are the same, we have (68) Equalities (67) and (68) imply the equality Consequently, (X, d) and (X, ρ) are combinatorially similar by Lemma 5.8.For the case when (X, d) and (X, ρ) are pseudorectangles the validity of (ii) ⇒ (i) can be proved similarly if apply Theorem 5.6 instead of Theorem 5.3 and In the next proposition we denote by c the cardinality of the continuum.
Proposition 5.10.Let X be a nonempty set, let ≡ be an equivalence relation on X and Q = {X j : j ∈ J} be the partition of X corresponding to ≡.Then the following statements hold: (i) The inequality |J| c holds if and only if there is a strongly rigid pseudometric space (X, d) such that is valid for all x, y ∈ X. (ii) The equality |J| = 4 holds if and only if there is a pseudorectangle (X, d) such that (69) is valid for all x, y ∈ X.
Proof.Statement (i) was proved in Theorem 4.13 of [3].Let us prove the validity of (ii).
Let |J| = 4 hold.Let us consider an injective mapping whenever B is a block of Q ⊗ 3 Q defined by equalities (48) − (50).Write ≡ 3 for the equivalence relation corresponding to the partition Q ⊗ 3 Q and denote by d the mapping where π and π⊗π are, respectively, the canonical projection of X on X/ ≡ and X 2 on X 2 / ≡ 3 .By Theorem 5.3, (X, d) is a pseudorectangle.(We note only that (71) implies the triangle inequality for d.)The definition of d and Lemma 5.1 imply that Q is the partition of X corresponding the equivalence relation 0(d) = .Let us consider a pseudorectangle (X, d) such that Q = {X j : j ∈ J} is a partition corresponding to the equivalence relation 0(d)  = .Then the equality |J| = 4 follows from Theorem 5.6.
(i 2 ) If (Y, ρ) ∈ CL, and Y = X, and the relations 0(d) = and 0(ρ) = are the same, then the identical mapping Id X : X → X is a combinatorial similarity of (X, d) and (Y, ρ).(i 3 ) Every nonempty subspace of (X, d) belongs to CL.Then exactly one from the following statements holds.
(ii 1 ) CL is the class of all strongly rigid pseudometric spaces.
(ii 2 ) CL is the class of all discrete pseudometric spaces.
(ii 3 ) CL is the union of the class of all pseudorectangles with the class of all strongly rigid pseudometric spaces (X, d) satisfying the inequality |d(X 2 )| 4.
Remark 5.13.The maximality of CL means that for every class CL o of nonempty pseudometric spaces, the inclusion CL o ⊇ CL implies the equality CL o = CL whenever CL o satisfies conditions (i 1 ) − (i 3 ) for every (X, d) ∈ CL o and every pseudometric space (Y, ρ).
Proof of Theorem 5.12.Let CL * be an arbitrary class of nonempty pseudometric spaces such that (i 1 ) − (i 3 ) are valid with CL = CL * for every (X, d) ∈ CL * and each pseudometric space (Y, ρ).
Let Φ : X/ 0 = → X/ 0 = be an arbitrary bijection of X/ 0 = .The function is a metric on X/ 0 = and Φ is an isometry of the metric spaces (X/ 0 = , δ d ) and (X/ 0 = , ρ Φ ).Since every isometry is a pseudoisometry, the membership (X/ 0 = , δ d ) ∈ CL * implies (X/ 0 = , ρ Φ ) ∈ CL * by condition (i 2 ).Furthermore, we have for all a, b ∈ X/ 0 = , because ρ is a metric on X/ 0 = .Thus, the relations 0(δ d ) = and 0(ρ Φ ) = are the same.Consequently, by condition (i 2 ), the identical mapping of X/ 0 = is a combinatorial similarity of (X/ 0 = , δ d ) and (X/ 0 = , ρ Φ ).Now using (72) and Definition 2.2, we obtain that Φ is a combinatorial self-similarity of (X/ 0 = , δ d ).Since Φ is an arbitrary element of Sym(X/ 0 = ), the equality holds.Consequently, by Theorem 4.4, at least one from the following statements is valid: For convenience we denote by CL st and CL di the classes of all strongly rigid pseudometroc spaces and all discrete ones respectively.In addition, write CL pr for the class of all pseudorectangles and CL 4 st for the class of all (X, d) ∈ CL st satisfying the inequality |d(X 2 )| 4.
Suppose that (Y, ρ) ∈ CL * met such that |Y | = 4 but (Y, ρ) is neither discrete nor strongly rigid.Then (73) implies that (Y, ρ) is a pseudorectangle.Then, using (i 2 ) and Proposition 5.9, we obtain that every (Y 1 , ρ 1 ) ∈ CL * met with |Y 1 | = 4 is also a pseudorectangle.Since every three-point subspace of (Y, ρ) is strongly rigid, we can prove that all (Z, d) ∈ CL * met with |Z| = 3 are also strongly rigid.Thus, CL * met ⊆ CL pr ∪ CL 4 st .To complete the proof that at least one inclusion from (76) holds it suffices to show that (77) holds if CL * met contains a metric space (Y, ρ) with |Y | 5.The last inequality implies that (Y, ρ) is not a pseudorectangle.Hence, by (73), (Y, ρ) is either discrete or strongly rigid.
If (Y, ρ) is discrete (strongly rigid), then arguing as above, we obtain that every (Y Thus, (78) implies that CL o = CL st , i.e.CL st is maximal.
The maximality of CL di and CL pr ∪ CL 4 st can be proved similarly.✷ Corollary 5.14.Let (Z, l) be a nonempty pseudometric space.Then the equality Cs(Z/ 0 = , δ l ) = Sym(Z/ 0 = ) holds, if and only if there is a class CL of nonempty pseudometric spaces, which satisfies (Z, l) ∈ CL and conditions (i 1 ) − (i 3 ) of Theorem 5.12 for every (X, d) ∈ CL and each nonempty pseudometric space (Y, ρ).
In connection with Theorem 5.12, the following question naturally arises.Are conditions (i 1 ) − (i 3 ) independent of each other?Funding Viktoriia Bilet was partially supported by the Grant EFDS-FL2-08 of the found The European Federation of Academies of Sciences and Humanities (ALLEA) and by a grant from the Simons Foundation (Award 1160640, Presidential Discretionary-Ukraine Support Grants).
Oleksiy Dovgoshey was supported by Finnish Society of Sciences and Letters.
u and y = v) or (x = v and y = u).

Corollary 3 . 8 .
Let d and ρ be discrete pseudometrics on a set X. Then the following statements are equivalent:(i) The pseudometric spaces (X, d) and (X, ρ) are combinatorially similar.(ii) The binary relation0(d)