Combinatorial properties of ultrametrics and generalized ultrametrics
aa r X i v : . [ m a t h . M G ] A ug COMBINATORIAL PROPERTIES OF ULTRAMETRICSAND GENERALIZED ULTRAMETRICS
OLEKSIY DOVGOSHEY
Abstract.
Let X , Y be sets and let Φ , Ψ be mappings withdomains X and Y respectively. We say that Φ and Ψ are com-binatorially similar if there are bijections f : Φ( X ) → Ψ( Y ) and g : Y → X such that Ψ( x, y ) = f (Φ( g ( x ) , g ( y ))) for all x , y ∈ Y .Conditions under which a given mapping is combinatorially similarto an ultrametric or a pseudoultrametric are found. Combinato-rial characterizations are also obtained for poset-valued ultrametricdistances recently defined by Priess-Crampe and Ribenboim. Introduction
Recall some definitions from the theory of metric spaces. Let X bea set, let X be the Cartesian square of X , X = X × X = {h x, y i : x, y ∈ X } , and let R + = [0 , ∞ ) . Definition 1.1. A metric on X is a function d : X → R + such thatfor all x , y , z ∈ X : ( i ) d ( x, y ) = 0 if and only if x = y , the positive property ; ( ii ) d ( x, y ) = d ( y, x ) , the symmetric property ; ( iii ) d ( x, y ) ≤ d ( x, z ) + d ( z, y ) , the triangle inequality .A metric d : X → R + is an ultrametric on X if ( iv ) d ( x, y ) ≤ max { d ( x, z ) , d ( z, y ) } holds for all x , y , z ∈ X .Inequality ( iv ) is often called the strong triangle inequality .The theory of ultrametric spaces is closely connected with various in-vestigations in mathematics, physics, linguistics, psychology and com-puter science. Different properties of ultrametrics have been studiedin [3–5,8,10–13,20–23,25–33,35,37–39,42,46–51,51,53,54,61,62,68–70]. Mathematics Subject Classification.
Primary 54E35, Secondary 06A05,06A06.
Key words and phrases. ultrametric, generalized ultrametric, equivalence rela-tion, poset, totally ordered set, isotone mapping.
An useful generalization of the concept of ultrametric is the conceptof pseudoultrametric and this is one of the main objects of our researchbelow.
Definition 1.2.
Let X be a set and let d : X → R + be a symmetricfunction such that d ( x, x ) = 0 holds for every x ∈ X . The function d is a pseudoultrametric ( pseudometric ) on X if it satisfies the strongtriangle inequality (triangle inequality).The strong triangle inequality also admits a natural generalizationfor poset-valued mappings.Let (Γ , ) be a partially ordered set with the smallest element γ and let X be a nonempty set. Definition 1.3.
A mapping d : X → Γ is an ultrametric distance , ifthe following conditions hold for all x , y , z ∈ X and γ ∈ Γ . ( i ) d ( x, y ) = γ if and only if x = y . ( ii ) d ( x, y ) = d ( y, x ) . ( iii ) If d ( x, y ) γ and d ( y, z ) γ , then d ( x, z ) γ .The ultrametric distances were introduced by Priess-Crampe andRibenboim [57] and studied in [58, 59, 63, 64]. This generalization ofultrametrics has some interesting applications to logic programming,computational logic and domain theory [44, 60, 66].Let us recall now the definition of combinatorial similarity. In whatfollows we will denote by F ( A ) the range of a mapping F : A → B , F ( A ) = { F ( x ) : x ∈ A } . Definition 1.4 ([16]) . Let X , Y be nonempty sets and let Φ , Ψ bemappings with the domains X and Y , respectively. The mapping Φ is combinatorially similar to Ψ if there are bijections f : Φ( X ) → Ψ( Y ) and g : Y → X such that(1.1) Ψ( x, y ) = f (Φ( g ( x ) , g ( y ))) holds for all x , y ∈ Y . In this case, we say that g : Y → X is a combinatorial similarity for the mappings Ψ and Φ .Equality (1.1) means that the diagram X Y Φ( X ) Ψ( Y ) g ⊗ gf Φ Ψ is commutative, where we understand the mapping g ⊗ g as ( g ⊗ g )( h y , y i ) := h g ( y ) , g ( y ) i OMBINATORIAL PROPERTIES 3 for h y , y i ∈ Y .Some characterizations of mappings which are combinatorially sim-ilar to pseudometrics, strongly rigid pseudometrics and discrete pseu-dometrics were obtained in [16]. The present paper deals with com-binatorial properties of ultrametrics and generalized ultrametrics andthis can be seen as a further development of research begun in [16, 19].The paper is organized as follows.In Section 2 we introduce the notions of strongly consistent mappingsand a -coherent mappings and show that these properties of mappingsare invariant w.r.t. combinatorial similarities, Proposition 2.4. Themain results of the section, Proposition 2.5 and Theorem 2.10, de-scribe a -coherent mappings in terms of binary relations defined onthe domains of these mappings. An important special case of combi-natorial similarities, the so-called weak similarities, are introduced inDefinition 2.12 at the end of the section.In Section 3, starting from the characterization of mappings whichare combinatorially similar to pseudometrics, we prove Theorem 3.10,a characterization of mappings which are combinatorially similar topseudoultrametrics with at most countable range. The correspond-ing results for ultrametrics are given in Corollary 3.11. A basic forour goals subclass of Priess-Crampe and Ribemboim ultrametric dis-tances, the Q -ultrametrics an related them Q -pseudoultrametrics,are introduced in Definition 3.14. In Proposition 4.3 we show that Q -pseudoultrametrics are a -coherent. The main result of the sec-tion is Theorem 3.18 which gives us the necessary and sufficient con-dition under which a given mapping is combinatorially similar to some Q -pseudoultrametric. Proposition 3.24 and Corollary 3.25 expand on Q -pseudoultrametrics the characterization of ultrametric-preservingfunctions obtained recently by Pongsriiam and Termwuttipong.Section 4 mainly describes the interrelations between combinatorialand weak similarities of Q -pseudoultrametrics. First of all, in Defini-tion 4.1, we expand the notion of weak similarity from usual pseudo-ultrametrics to Q -pseudoultrametrics. Proposition 4.3 claims that,for all Q -pseudoultrametrics, every weak similarity is a combinatorialsimilarity (but not conversely in general). The orders Q , for whichthe weak similarities and the combinatorial similarities are the same(for the corresponding Q -pseudoultrametrics) are described in Theo-rem 4.4. In Proposition 4.7, for every totally ordered set ( Q, Q ) (whichcontains a smallest element) we construct a Q -ultrametric satisfyingconditions of Theorem 4.4. Using this result in Proposition 4.11 wefound a metric d ∗ , defined on a set X with | X | = 2 ℵ , such that d ∗ OLEKSIY DOVGOSHEY is not combinatorially similar to any ultrametric but, for every count-able X ⊆ X , the restriction d ∗ on X is combinatorially similar to anultrametric. The mappings which are combinatorially similar to Q -pseudoultrametrics are described in Theorems 4.15, 4.18 and 4.20 forthe case of totally ordered ( Q, Q ) satisfying the distinct universal andtopological restrictions. The final results of the paper, Theorem 4.21and Corollary 4.22, give a kind of necessary and sufficient conditionsunder which a given mapping is combinatorially similar to a pseudoul-trametric or, respectively, to an ultrametric.2. Consistency with equivalence relations
Let X be a set. A binary relation on X is a subset of the Cartesiansquare X . A relation R ⊆ X is an equivalence relation on X if thefollowing conditions hold for all x , y , z ∈ X : ( i ) h x, x i ∈ R , the reflexive property; ( ii ) ( h x, y i ∈ R ) ⇔ ( h y, x i ∈ R ) , the symmetric property; ( iii ) (( h x, y i ∈ R ) and ( h y, z i ∈ R )) ⇒ ( h x, z i ∈ R ) , the transitive property.Let R be an equivalence relation on X . A mapping F : X → X is consistent with R if the implication (cid:0) h x , x i ∈ R and h x , x i ∈ R (cid:1) ⇒ (cid:0) h F ( x , x ) , F ( x , x ) i ∈ R (cid:1) is valid for all x , x , x , x ∈ X (see [45, p. 78]). Similarly, we willsay that a mapping Φ : X → Y is strongly consistent with R if theimplication(2.1) (cid:0) h x , x i ∈ R and h x , x i ∈ R (cid:1) ⇒ (cid:0) Φ( x , x ) = Φ( x , x ) (cid:1) is valid for all x , x , x , x ∈ X . Remark . Let R be an equivalence relation on a set X . Then everystrongly consistent with R mapping Φ : X → X is consistent with R .The converse statement holds if and only if R is the diagonal of X , R = ∆ X = {h x, x i : x ∈ X } . Definition 2.2.
Let X be a nonempty set, let Φ be a mapping with dom Φ = X and let a ∈ Φ( X ) . The mapping Φ is a - coherent if Φ is strongly consistent with the fiber Φ − ( a ) := {h x, y i : Φ( x, y ) = a } . Remark . In particular, if Φ is a -coherent, then Φ − ( a ) is an equiv-alence relation on X . OMBINATORIAL PROPERTIES 5
The following proposition claims that the properties to be stronglyconsistent and to be coherent are invariant w.r.t. combinatorial simi-larities.
Proposition 2.4.
Let X , Y be nonempty sets, let Φ , Ψ be combina-torially similar mappings with dom Φ = X and dom Ψ = Y and thecommutative diagram X Y Φ( X ) Ψ( Y ) g ⊗ gf Φ Ψ . If Φ is strongly consistent with an equivalence relation R X on X , then Ψ is strongly consistent with an equivalence relation R Y on Y satisfying ( h x, y i ∈ R Y ) ⇔ ( h g ( x ) , g ( y ) i ∈ R X ) for every h x, y i ∈ Y . In addition, if Φ is a -coherent for a ∈ Φ( X ) ,then Ψ is f ( a ) -coherent. The proof is straightforward and we omit it here.Let X be a set and let R and R be binary relations on X . Recallthat a composition of binary relations R and R is a binary relation R ◦ R ⊆ X for which h x, y i ∈ R ◦ R holds if and only if there is z ∈ X such that h x, z i ∈ R and h z, y i ∈ R .Using the notion of binary relations composition we can reformulateDefinition 2.2 as follows. Proposition 2.5.
Let X be a nonempty set, Φ be a mappings with dom Φ = X and let a ∈ Φ( X ) . Then Φ is a -coherent if and only ifthe fiber R = Φ − ( a ) is an equivalence relation on X and the equality (2.2) Φ − ( b ) = R ◦ Φ − ( b ) ◦ R holds for every b ∈ Φ( X ) .Proof. It suffices to show that Φ is strongly consistent with R if andonly if equality (2.2) holds for every b ∈ Φ( X ) . Let b ∈ Φ( X ) and(2.2) hold. Suppose h x , x i ∈ X such that Φ( x , x ) = b. If h x , x i ∈ R , h x , x i ∈ Φ − ( b ) and h x , x i ∈ R , then from thedefinition of the composition ◦ we obtain h x , x i ∈ R ◦ Φ − ( b ) ◦ R OLEKSIY DOVGOSHEY that implies h x , x i ∈ Φ − ( b ) by equality (2.2). Thus, the implica-tion (2.1) is valid.Conversely, suppose that Φ is strongly consistent with R . Then (2.1)implies the inclusion(2.3) R ◦ Φ − ( b ) ◦ R ⊆ Φ − ( b ) for every b ∈ Φ( X ) . Since R is reflexive, the converse inclusion is alsovalid. Equality (2.2) follows. (cid:3) Corollary 2.6.
Let X be a nonempty set, let Φ be a symmetric map-ping with dom Φ = X and let a ∈ Φ( X ) . Suppose R := Φ − ( a ) is an equivalence relation on X . Then the following conditions areequivalent. ( i ) Φ is a -coherent. ( ii ) Φ − ( b ) = R ◦ Φ − ( b ) ◦ R holds for every b ∈ Φ( X ) . ( iii ) Φ − ( b ) = R ◦ Φ − ( b ) holds for every b ∈ Φ( X ) . ( iv ) Φ − ( b ) = Φ − ( b ) ◦ R holds for every b ∈ Φ( X ) . ( v ) For every b ∈ Φ( X ) , at least one of the equalities Φ − ( b ) = R ◦ Φ − ( b ) , Φ − ( b ) = Φ − ( b ) ◦ R holds.Proof. In what follows, for every b ∈ Φ( X ) , we write R b = Φ − ( b ) and,for every A ⊆ X , define the inverse binary relation A T by the rule: • the membership h x, y i ∈ A T holds if and only if h y, x i ∈ A .Suppose ( v ) is valid and we have(2.4) R b = R b ◦ R. It is trivial that a binary relation A is symmetric if and only if we have A T = A . Furthermore, the equality ( C ◦ B ) T = B T ◦ C T holds for all binary relations B and C defined on the one and the sameset (see, for example, [36, p. 15]). Consequently, from (2.4) it followsthat R b = ( R b ) T = ( R b ◦ R ) T = R T ◦ R Tb = R ◦ R b = R ◦ ( R b ◦ R ) = R ◦ R b ◦ R. Similarly, from R b = R ◦ R b follows R b = R ◦ R b ◦ R . Thus, theimplication ( v ) ⇒ ( ii ) is valid.If ( ii ) holds, then we have R b = R ◦ R b ◦ R OMBINATORIAL PROPERTIES 7 for every b ∈ Φ( X ) . Since R is an equivalence relation, the equality R ◦ R = R holds. Consequently, R b = ( R ◦ R ) ◦ R b ◦ R = R ◦ ( R ◦ R b ◦ R ) = R ◦ R b . Thus, ( ii ) implies ( iii ) . Analogously, ( ii ) implies ( iv ) . The implications ( iii ) ⇒ ( v ) and ( iv ) ⇒ ( v ) are evidently valid. To complete the proofwe recall that ( i ) and ( ii ) are equivalent by Proposition 2.5. (cid:3) Let X be a nonempty set and P = { X j : j ∈ J } be a set of nonemptysubsets of X . Then P is a partition of X with the blocks X j if [ j ∈ J X j = X and X j ∩ X j = ∅ holds for all distinct j , j ∈ J .There exists the well-known, one-to-one correspondence between theequivalence relations and partitions.If R is an equivalence relation on X , then an equivalence class is asubset [ a ] R of X having the form(2.5) [ a ] R = { x ∈ X : h x, a i ∈ R } , a ∈ X. The quotient set of X w.r.t. R is the set of all equivalence classes [ a ] R , a ∈ X . Proposition 2.7.
Let X be a nonempty set. If P = { X j : j ∈ J } is apartition of X and R P is a binary relation on X defined as h x, y i ∈ R P if and only if ∃ j ∈ J such that x ∈ X j and y ∈ X j ,then R P is an equivalence relation on X with the equivalence classes X j . Conversely, if R is an equivalence relation on X , then the set P R of all distinct equivalence classes [ a ] R is a partition of X with the blocks [ a ] R . For the proof, see, for example, [45, Chapter II, § 5].
Lemma 2.8 ([41, p. 9]) . Let X be a nonempty set. If R is an equiv-alence relation on X and P R = { X j : j ∈ J } is the correspondingpartition of X , then the equality R = [ j ∈ J X j holds. For every partition P = { X j : j ∈ J } of a nonempty set X we definea partition P ⊗ P of X by the rule: OLEKSIY DOVGOSHEY • A subset B of X is a block of P ⊗ P if and only if either B = [ j ∈ J X j or there are distinct j , j ∈ J such that B = X j × X j . Definition 2.9.
Let X be a nonempty set and let P and P be par-titions of X . The partition P is finer than the partition P if theinclusion [ x ] R P ⊆ [ x ] R P holds for every x ∈ X , where R P and R P are equivalence relationscorresponding to P and P respectively.If P is finer than P , then we say that P is a refinement of P .The following proposition gives us a new characterization of a -coherent mappings. Theorem 2.10.
Let X be a nonempty set, Φ be a mapping with dom Φ = X and let a ∈ Φ( X ) . Then Φ is a -coherent if and only ifthe fiber R := Φ − ( a ) is an equivalence relation on X and the partition P R ⊗ P R of X is arefinement of the partition P Φ − := { Φ − ( b ) : b ∈ Φ( X ) } , where P R isa partition of X whose blocks are the equivalence classes of R .Proof. Let Φ be a -coherent. Then, by Definition 2.2, R is an equiva-lence relation on X . We claim that P R ⊗ P R is a refinement P Φ − . Itsuffices to show that for every block B of P R ⊗ P R there is b ∈ Φ( X ) such that(2.6) B ⊆ Φ − ( b ) . Suppose that(2.7) B = [ j ∈ J X j , where X j , j ∈ J , are the blocks of the partition corresponding to theequivalence relation Φ − ( a ) on X . By Lemma 2.8, we have the equality [ j ∈ J X j = Φ − ( a ) . OMBINATORIAL PROPERTIES 9
The last equality and (2.7) imply (2.6) with b = a . If B is a block of P R ⊗ P R but (2.7) does not hold, then there are two distinct j , j ∈ J such that(2.8) B = X j × X j . Let x ∈ X j and x ∈ X j and let b ∈ Φ( X ) such that(2.9) h x , x i ∈ Φ − ( b ) . We must show that(2.10) X j × X j ⊆ Φ − ( b ) . It follows from Proposition 2.5 and Lemma 2.8 that(2.11) Φ − ( b ) = [ j ∈ J X j ! ◦ Φ − ( b ) ◦ [ j ∈ J X j ! holds. Inclusion (2.10) holds if, for every x ∈ X j and y ∈ X j , we have h x, y i ∈ Φ − ( b ) . Using (2.11) we obtain(2.12) Φ − ( b ) ⊇ X j ◦ Φ − ( b ) ◦ X j . Since h x, x i ∈ X j and h x , x i ∈ Φ − ( b ) and h x , y i ∈ X j , thedefinition of composition ◦ and (2.12) imply h x, y i ∈ Φ − ( b ) . Thus, P R ⊗ P R is a refinement of P Φ − if Φ is a -coherent.Conversely, suppose that R = Φ − ( a ) is an equivalence relation on X and P R ⊗ P R is a finer than P Φ − . By Proposition 2.5, the mapping Φ is a -coherent if and only if the equality R ◦ Φ − ( b ) ◦ R = Φ − ( b ) holds for every b ∈ Φ( X ) . The reflexivity of R implies that R ◦ Φ − ( b ) ◦ R ⊇ Φ − ( b ) . Consequently, to complete the proof it suffices to show that(2.13) R ◦ Φ − ( b ) ◦ R ⊆ Φ − ( b ) holds for every b ∈ Φ( X ) . Inclusion (2.13) holds if and only if(2.14) R ◦ {h x, y i} ◦ R ⊆ Φ − ( b ) holds for every h x, y i ∈ Φ − ( b ) , where {h x, y i} is the one-point subsetof X consisting the point h x, y i only. A simple calculation shows that(2.15) B = R ◦ B ◦ R holds for every block B of the partition P R ⊗ P R . Since P R ⊗ P R is arefinement of P Φ − , equality (2.15) implies (2.14) for h x, y i ∈ B . (cid:3) Let us consider now some examples.
Proposition 2.11.
Let X be a nonempty set and let d : X → R + bea pseudoultrametric on X . Then d − (0) is an equivalence relation on X and d is -coherent. This proposition is a corollary of the corresponding result for pseu-dometrics [41, Ch. 4, Th. 15].
Definition 2.12.
Let ( X , d ) and ( X , d ) be pseudoultrametricspaces. A bijection Φ : X → X is a weak similarity if there is astrictly increasing bijective function f : d ( X ) → d ( X ) such thatthe equality(2.16) d ( x, y ) = f ( d (Φ( x ) , Φ( y ))) holds for all x , y ∈ X . Remark . The weak similarities of semimetric spaces and ultra-metric ones were studied in [24] and [53]. See also [43] and referencestherein for some results related to weak similarities of subsets of Eu-clidean finite-dimensional spaces.
Proposition 2.14.
Let ( X , d ) and ( X , d ) be pseudoultrametricspaces and Φ : X → X be a weak similarity. Then Φ is a combi-natorial similarity for the pseudoultrametrics d and d .Proof. It follows directly from Definition 2.12 and Definition 1.4. (cid:3) Combinatorial similarity for generalized ultrametrics
First of all, we recall a combinatorial characterization of arbitrarypseudometric.
Theorem 3.1 ([16]) . Let X be a nonempty set. The following condi-tions are equivalent for every mapping Φ with dom Φ = X . ( i ) Φ is combinatorially similar to a pseudometric. ( ii ) Φ is symmetric, and | Φ( X ) | ℵ , and there is a ∈ Φ( X ) such that Φ is a -coherent. Corollary 3.2 ([16]) . Let X be a nonempty set and let Φ be a mappingwith dom Φ = X . Then Φ is combinatorially similar to a metric if andonly if Φ is symmetric, and | Φ( X ) | ℵ , and there is a ∈ Φ( X ) such that Φ − ( a ) = ∆ X , where ∆ X is the diagonal of X . Consequently, if a mapping Φ , with dom Φ = X , is combinatori-ally similar to a pseudoultrametric, then it satisfies condition ( ii ) ofTheorem 3.1. OMBINATORIAL PROPERTIES 11
Another necessary condition for combinatorial similarity of Φ to apseudoultrametric follows from the fact that • all triangles are isosceles in every pseudoultrametric space.This fact can be written in the form. Lemma 3.3.
Let X be a nonempty set and let Φ be a mapping with dom Φ = X . If Φ is combinatorially similar to a pseudoultrametric,then, ( i ) for every triple h x , x , x i of points of X , there is a permuta-tion (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) . The following example shows that condition ( i ) is not sufficient forexistence of a pseudoultrametric d which is combinatorially similar to Φ , even Φ is a metric. Example 3.4.
Let X = { x , x , x , x } and let ρ : X → R + be asymmetric mapping defined as(3.1) ρ ( x, y ) = if x = y, π if { x, y } = { x , x } or { x, y } = { x , x } ,π otherwise . It is easy to see that ρ is a metric on X such that every triangle isisosceles in ( X, ρ ) (see Figure 1). Suppose ρ is combinatorially similarto some pseudoultrametric d : Y → R + . Then, by Definition 1.4, thereare bijections f : ρ ( X ) → d ( Y ) and g : Y → X such that d ( x, y ) = f ( ρ ( g ( x ) , g ( y ))) for all x , y ∈ Y . The last equality and (3.1) imply d ( g − ( x ) , g − ( x )) = d ( g − ( x ) , g − ( x )) = f (cid:16) π (cid:17) and d ( g − ( x ) , g − ( x )) = d ( g − ( x ) , g − ( x )) = f ( π ) . Using these equalities and the strong triangle inequality (for the triples h g − ( x ) , g − ( x ) , g − ( x ) i and h g − ( x ) , g − ( x ) , g − ( x ) i ) we obtain f (cid:16) π (cid:17) > f ( π ) and f (cid:16) π (cid:17) f ( π ) . Thus, f (cid:0) π (cid:1) = f ( π ) holds, contrary to the bijectivity of f . x x x x Figure 1.
The metric space ( X, ρ ) is (up to isometry)a subspace of the metric space L consisting of the threerays −−→ x x , −−→ x x , −−→ x x and a unit circle (a circle with theradius ) passing through x , x and x if we consider L endowed with the shortest path metric.We want to describe the mappings which are combinatorially similarto pseudoultrametrics. For this goal we recall some definitions.Let γ be a binary relation on a set X . We will write γ = γ and γ n +1 = γ n ◦ γ for every integer n > . The transitive closure γ t of γ isthe relation(3.2) γ t := ∞ [ n =1 γ n . For every β ⊆ X , the transitive closure β t is transitive and theinclusion β ⊆ β t holds. Moreover, if τ ⊆ X is an arbitrary transitivebinary relation for which β ⊆ τ , then we also have β t ⊆ τ , i.e., β t isthe smallest transitive binary relation containing β .Recall that a reflexive and transitive binary relation Y on a set Y is a partial order on Y if, for all x , y ∈ Y , we have the antisymmetricproperty , (cid:0) h x, y i ∈ Y and h y, x i ∈ Y (cid:1) ⇒ ( x = y ) . In what follows we use the formula x y instead of h x, y i ∈ andwrite x ≺ y instead of x y and x = y. Let Y be a partial order on a set Y . A pair ( Y, Y ) is called to bea poset (a partially ordered set). A poset ( Y, Y ) is linear (= totallyordered ) if, for all y , y ∈ Y , we have y Y y or y Y y . OMBINATORIAL PROPERTIES 13
Definition 3.5.
Let ( Q, Q ) and ( L, L ) be posets. A mapping f : Q → L is isotone if, for all q , q ∈ Q , we have ( q Q q ) ⇒ ( f ( q ) L f ( q )) . Let
Φ : X → Y be an isotone mapping of posets ( X, X ) and ( Y, Y ) .If Φ is bijective and the inverse mapping Φ − : Y → X is also isotone,then we say that ( X, X ) and ( Y, Y ) are isomorphic and Φ is an( order ) isomorphism .If ( Y, Y ) is a poset, and Y ⊆ Y , and Y is a partial order on Y such that, for all x , y ∈ Y , ( x Y y ) ⇔ ( x Y y ) , then we say that ( Y , Y ) is a subposet of the poset ( Y, Y ) .Write Q + for the set of all nonnegative rational numbers, Q + = Q ∩ [0 , + ∞ ) , and let be the usual ordering on Q + . Lemma 3.6 (Cantor) . Let ( X, X ) be a totally ordered set and let | X | ℵ hold. Then ( X, X ) is isomorphic to a subposet of ( Q + , ) . The proof can be obtained directly from the classical Cantor’s results(see, for example, [65], Chapter 2, Theorem 2.6 and Theorem 2.8).We will also use the following Szpilrajn Theorem.
Lemma 3.7 (Szpilrajn) . Let ( X, X ) be a poset. Then there is a linearorder on X such that X ⊆ . Informally speaking it means that each partial order on a set can beextended to a linear order on the same set.
Remark . This result was obtained by Edward Szpilrajn in [67].Interesting reviews of Szpilrajn-type theorems can be found in [2] and[7].Let X be a nonempty set and let Φ be a symmetric mapping with dom Φ = X and let Y := Φ( X ) . Let us define a binary relation u Φ by the rule: h y , y i ∈ u Φ if and only if h y , y i ∈ Y and there are x , x , x ∈ X such that(3.3) y = Φ( x , x ) and y = Φ( x , x ) = Φ( x , x ) . Example 3.9.
Let ( X, d ) be a nonempty ultrametric space. Recallthat a subset B of X is a (closed) ball if there are x ∗ ∈ X and r ∗ ∈ R + such that B = { x ∈ X : d ( x, x ∗ ) r ∗ } . The diameter of B , we denote it by diam( B ) , is defined as diam( B ) := sup { d ( x, y ) : x, y ∈ B } . The following statements are equivalent for every h r , r i ∈ R + × R + . • h r , r i ∈ u d . • There are some balls B and B in ( X, d ) such that B ⊆ B ,and r = diam( B ) , and r = diam( B ) . • There are some balls B and B in ( X, d ) such that B ∩ B = ∅ , and r = diam( B ) , r = diam( B ) , and r r .The interchangeability of these conditions is easy to justify using theknown properties of balls in ultrametric spaces (see, for example,Proposition 1.2 and Proposition 1.6 in [17]). Theorem 3.10.
Let X be a nonempty set and let Φ be a mapping with dom Φ = X and | Φ( X ) | ℵ . Then the following conditions areequivalent. ( i ) Φ is combinatorially similar to a pseudoultrametric d : X → R + with d ( X ) ⊆ Q + . ( ii ) Φ is combinatorially similar to a pseudoultrametric. ( iii ) The mapping Φ is symmetric, and the transitive closure u t Φ ofthe binary relation u Φ is antisymmetric, and Φ is a -coherentfor a point a ∈ Φ( X ) , and, for every triple h x , x , x i ofpoints of X , there is a permutation (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) .Proof. ( i ) ⇒ ( ii ) . This is trivially valid. ( ii ) ⇒ ( iii ) . Suppose Φ is combinatorially similar to a pseudoul-trametric. Then Φ also is combinatorially similar to a pseudometric.Consequently, by Theorem 3.1, Φ is symmetric and there is a ∈ Φ( X ) such that Φ is a -coherent. If h x , x , x i is an arbitrary triple of pointsof X , then, by Lemma 3.3, there is a permutation (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) . To complete the proof of validityof ( ii ) ⇒ ( iii ) it suffices to show that the transitive closure u t Φ of thebinary relation u Φ ⊆ Y , Y = Φ( X ) , is antisymmetric. Suppose contrary that there are distinct y , y ∈ Y such that h y , y i ∈ u t Φ and h y , y i ∈ u t Φ . The definition of the OMBINATORIAL PROPERTIES 15 transitive closure (see (3.2)) and the definition of the composition ofbinary relations imply that there are a positive integer n and somepoints y ∗ , y ∗ , . . . , y ∗ n +1 ∈ Y with(3.4) y ∗ = y and y ∗ n +1 = y and h y ∗ i , y ∗ i +1 i ∈ u Φ for i = 1 , . . . , n . Since Φ is combinatorially similar to a pseudoultra-metric d : Z → R + , there are bijections g : Z → Y and f : Φ( X ) → d ( Z ) satisfying d ( z , z ) = f (Φ( g ( z ) , g ( z ))) for all z , z ∈ Z . Consequently, d ( g − ( x ) , g − ( x )) = f (Φ( x , x )) holds for all x , x ∈ X . As in Example 3.4, the last equality, (3.3),(3.4), and the strong triangle inequality imply f ( y ) = f ( y ∗ ) > f ( y ∗ ) > . . . > f ( y ∗ n +1 ) = f ( y ) . Thus, the inequality f ( y ) > f ( y ) holds. Similarly, we can obtain theinequality f ( y ) > f ( y ) .Consequently, the equality f ( y ) = f ( y ) holds, that contradicts thebijectivity of f . ( iii ) ⇒ ( i ) . Suppose Φ satisfies condition ( iii ) . Let us define a binaryrelation on Y = Φ( X ) as(3.5) := u t Φ ∪ ∆ Y , where ∆ Y = {h y, y i : y ∈ Y } . We claim that is a partial order on Y . Indeed, (3.5) implies that is reflexive. By condition ( iii ) , thetransitive closure u t Φ is antisymmetric. From this and (3.5) it followsthat is also antisymmetric. Moreover, using the transitivity of u t Φ we obtain ( u t Φ ∪ ∆ Y ) = ( u t Φ ◦ u t Φ ) ∪ ( u t Φ ◦ ∆ Y ) ∪ (∆ Y ◦ u t Φ ) ∪ (∆ Y ◦ ∆ Y ) ⊆ u t Φ ∪ ∆ Y . Consequently, is transitive. Thus, is a partial order as required.By condition ( iii ) , Φ is a -coherent. We will show that a is thesmallest element of the poset ( Y, ) .Let y be an arbitrary point of Y . Then there are x , x ∈ X suchthat y = Φ( x , x ) . The mapping Φ is symmetric. Thus,(3.6) Φ( x , x ) = Φ( x , x ) holds. Since Φ is a -coherent, we have(3.7) Φ( x , x ) = a . Using (3.6), (3.7) and the definition of u Φ we obtain h a , y i ∈ u Φ forevery y ∈ Y , as required.Write for the intersection with the set Y , where Y = { y ∈ Y : y = a } . Then is a partial order on the set Y . By Lemma 3.7, there is alinear order ∗ on Y such that ⊆ ∗ . The inequality | Y | ℵ implies | Y | ℵ . Using Lemma 3.6 we canfind an injective mapping f ∗ : Y → Q + such that f ∗ ( a ) = 0 and ( y ∗ y ) ⇔ ( f ∗ ( y ) f ∗ ( y )) for all y , y ∈ Y . Then the function d : X → R + , d ( x , x ) = f ∗ (Φ( x , x )) , x , x ∈ X, is a pseudoultrametric on X and d ( X ) ⊆ Q + holds. Since the func-tion f ∗ is injective, the identical mapping X id −→ X is a combinatorialsimilarity. (cid:3) Using Theorem 3.10 and Corollary 3.2 we also obtain.
Corollary 3.11.
Let X be a nonempty set. The following conditionsare equivalent for every mapping Φ with dom Φ = X and | Φ( X ) | ℵ . ( i ) Φ is combinatorially similar to an ultrametric d : X → R + satisfying the inclusion d ( X ) ⊆ Q + . ( ii ) Φ is combinatorially similar to an ultrametric. ( iii ) Φ is symmetric, and the transitive closure u t Φ of the binaryrelation u Φ is antisymmetric, and the equality Φ − ( a ) = ∆ X holds for some a ∈ Φ( X ) , and, for every triple h x , x , x i of points of X , there is a permutation (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) . OMBINATORIAL PROPERTIES 17
Example 3.12.
A four-point metric space ( X, d ) is called a pseudo-linear quadruple (see [6] for instance) if, for a suitable enumeration ofpoints of X , we have(3.8) d ( x , x ) = d ( x , x ) = s, d ( x , x ) = d ( x , x ) = t,d ( x , x ) = d ( x , x ) = s + t, with some positive reals s and t . For a pseudolinear quadruple ( X, d ) ,Corollary 3.11 implies that the metric d : X → R + is combinatoriallysimilar to an ultrametric if and only if ( X, d ) is “equilateral”, i.e., (3.8)holds with s = t (see Figure 2). x x x x Figure 2.
Each equilateral, pseudolinear quadruple is(up to similarity) a subspace { x , x , x , x } of the unitcircle endowed with the shortest path metric. Remark . The pseudolinear quadruples appeared for the first timein the paper of Menger [52]. According to Menger, the pseudolinearquadruples are characterized as the metric spaces which are not iso-metric to any subset of R , but such that every triple of whose pointsembeds isometrically into R . There is also an elementary proof of thisfact [15]. It is interesting to note that the equilateral, pseudolinearquadruples are the “most non-Ptolemaic” metric spaces [14].For what follows we need a specification of the concept of ultrametricdistances introduced above in Definition 1.3. Definition 3.14.
Let ( Q, Q ) be a poset with a smallest element q and let X be a nonempty set. A mapping d : X → Q is a Q - pseudo-ultrametric if d is symmetric and d ( x, x ) = q holds for every x ∈ X and, in addition, for every triple h x , x , x i of points of X , there is apermutation (cid:18) x x x x i x i x i (cid:19) such that(3.9) d ( x i , x i ) Q d ( x i , x i ) and d ( x i , x i ) = d ( x i , x i ) . For Q -pseudoultrametric d , satisfying d ( x, y ) = q if and only if x = y ,we say that d is a Q - ultrametric .If there is no ambiguity in the choice of the order Q we write “ d isa Q -pseudoultrametric” instead of “ d is a Q -pseudoultrametric”. Remark . It is easy to prove that every ultrametric is a -ultra-metric for ( R + , ) . Moreover, every Q -ultrametric is an ultrametricdistance with the same ( Q, Q ) but not conversely (see, in particular,Example 3.26 at the end of the present section). For all totally orderedsets Q , the ultrametric distances coincide with Q -ultrametrics, andwith generalized ultrametrics defined by Priess-Crampe [56].The following proposition is an extension of Proposition 2.11 for thecase of arbitrary Q -pseudoultrametric. Proposition 3.16.
Let X be a nonempty set and ( Q, Q ) be a posetwith the smallest element q and let d : X → Q be a Q -pseudoultra-metric on X . Then d − ( q ) is an equivalence relation on X and themapping d is q -coherent.Proof. It follows directly from Definition 3.14 that d − ( q ) is reflexive.To prove that d − ( q ) is symmetric it suffices to note that the mapping d : X → Q is symmetric, because, for each mapping Φ with dom Φ = X , Φ is symmetric if and only if Φ − ( b ) is a symmetric binary relation forevery b ∈ Φ( X ) . Thus, d − ( q ) is an equivalence relation if and onlyif d − ( q ) is transitive.Let h x , x i and h x , x i belong to X and let(3.10) d ( x , x ) = d ( x , x ) = q . We claim that d ( x , x ) = q holds. Indeed, by Definition 3.14, thereis a permutation (cid:18) x x x x i x i x i (cid:19) such that (3.9) holds. From (3.10) and (3.9) it follows that(3.11) d ( x i , x i ) = d ( x i , x i ) = q . Using (3.9) again we see that (3.11) implies(3.12) d ( x i , x i ) Q q . Since q is the smallest element of ( Q, Q ) , inequality (3.12) implies(3.13) d ( x i , x i ) = q . OMBINATORIAL PROPERTIES 19
The equality d ( x , x ) = q follows from (3.13) and (3.11). Thus, d − ( q ) is transitive.Now we need to prove that d is q -coherent. The mapping d issymmetric. Hence, by Corollary 2.6, it suffices to show that(3.14) d − ( q ) = d − ( q ) ◦ d − ( q ) for every q ∈ d ( X ) . Let q ∈ d ( X ) . We have d − ( q ) ⊆ d − ( q ) ◦ d − ( q ) , because d − ( q ) is reflexive. The converse inclusion(3.15) d − ( q ) ⊇ d − ( q ) ◦ d − ( q ) holds if and only if, for all x , x , x ∈ X , we have(3.16) h x , x i ∈ d − ( q ) whenever h x , x i ∈ d − ( q ) and h x , x i ∈ d − ( q ) . If q = q , then(3.15) holds, since d − ( q ) is an equivalence relation. Suppose q = q . Write q := d ( x , x ) . If q = q , then (3.16) follows from h x , x i ∈ d − ( q ) . Consequently, if (3.16) is false, then we have(3.17) q = q = q . The equality q = q implies(3.18) h x , x i ∈ d − ( q ) , because d − ( q ) is transitive. From (3.18) and (3.16) follows q = q ,contrary to (3.17). Thus, q , q and q are pairwise distinct, thatcontradicts (3.9). (cid:3) Corollary 3.17.
Let X be a nonempty set and ( Q, Q ) be a poset andlet d : X → Q be a Q -pseudoultrametric ( Q -ultrametric) on X . Thenthe following statements are valid. ( i ) If | d ( X ) | ℵ holds, then d is combinatorially similar to anusual pseudometric (metric). ( ii ) If | d ( X ) | ℵ holds, then d is combinatorially similar to anusual pseudoultrametric (ultrametric).Proof. Suppose first that d is a Q -pseudoultrametric. ( i ) . If | d ( X ) | ℵ holds, then Definition 3.14 and Proposition 3.16imply condition ( ii ) of Theorem 3.1. Thus, ( i ) is valid by Theorem 3.1. ( ii ) . Analogously, using Definition 3.14 we can show that condi-tion ( iii ) of Theorem 3.10 is valid for Φ = d . Thus, ( ii ) follows fromTheorem 3.10.The case when d is a Q -ultrametric can be considered similarly. (cid:3) The next theorem is a partial generalization of Theorem 3.10.
Theorem 3.18.
Let X be a nonempty set and let Φ be a mapping with dom Φ = X . Then the following conditions are equivalent. ( i ) There is a totally ordered set Q such that Φ is combinatoriallysimilar to a Q -pseudoultrametric. ( ii ) There is a poset Q such that Φ is combinatorially similar to a Q -pseudoultrametric. ( iii ) The mapping Φ is symmetric, and the transitive closure u t Φ of the binary relation u Φ is antisymmetric, and there is a ∈ Φ( X ) for which Φ is a -coherent, and, for every triple h x , x , x i of points of X , there is a permutation (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) . ( iv ) There is b ∈ Φ( X ) such that Φ( x, x ) = b holds for every x ∈ X , and the binary relation (3.19) Φ := u t Φ ∪ ∆ Φ( X ) is a partial order on Φ( X ) , and b is the smallest element of (Φ( X ) , Φ ) , and Φ is a Φ -pseudoultrametric on X .Proof. The implication ( i ) ⇒ ( ii ) is trivially valid. The validity of ( ii ) ⇒ ( iii ) can be verified by repetition of the first part of the proof ofTheorem 3.10 with the replacement of the word “Theorem 3.1” by word“Proposition 3.16”. It should be noted that Lemma 3.3 remains validif Φ is combinatorially similar to an arbitrary Q -pseudoultrametric. ( iii ) ⇒ ( iv ) . Let ( iii ) hold. Then u t Φ is antisymmetric and tran-sitive. Consequently, the relation Φ is reflexive, antisymmetric andtransitive, i.e., Φ is a partial order on Φ( X ) . Since Φ is a -coherent,the equality Φ( x, x ) = a holds for every x ∈ X .The point a is the smallest element of (Φ( X ) , Φ ) if and only ifthe inequality(3.20) a Φ Φ( x, y ) holds for all x , y ∈ X . To prove (3.20) we consider the triple h y, x, y i and note that Φ( x, y ) = Φ( y, x ) . Consequently, h Φ( x, x ) , Φ( x, y ) i be-longs to u Φ . Now (3.20) follows from (3.19).By condition ( iii ) , Φ( x, x ) = a holds for every x ∈ X and, for everytriple h x , x , x i of points of X , there is a permutation (cid:18) x x x x i x i x i (cid:19) OMBINATORIAL PROPERTIES 21 such that Φ( x i , x i ) = Φ( x i , x i ) . The mapping Φ is symmetric.Hence, Φ is a Φ -pseudoultrametric on X as required. ( iv ) ⇒ ( i ) . Let ( iv ) hold. Then Φ is a Φ -pseudoultrametric. ByLemma 3.7 (Szpilrajn) the partial order Φ can be extended to a linearorder on Φ( X ) . It is easy to see that the smallest element a of (Φ( X ) , Φ ) is also the smallest element of (Φ( X ) , ) . Thus, Φ isalso a -pseudoultrametric. Condition ( i ) follows. (cid:3) Corollary 3.19.
Let X be a nonempty set and let Φ be a mapping with dom Φ = X . Then the following conditions are equivalent. ( i ) There is a totally ordered set Q such that Φ is combinatoriallysimilar to a Q -ultrametric. ( ii ) There is a poset Q such that Φ is combinatorially similar to a Q -ultrametric. ( iii ) The mapping Φ is symmetric, and the transitive closure u t Φ of the binary relation u Φ is antisymmetric, and there is a ∈ Φ( X ) for which Φ − ( a ) = ∆ X holds, and, for every triple h x , x , x i of points of X , there is a permutation (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) . ( iv ) There is b ∈ Φ( X ) such that Φ − ( b ) = ∆ X holds, and thebinary relation Φ := u t Φ ∪ ∆ Φ( X ) is a partial order on Φ( X ) , and b is the smallest element of (Φ( X ) , Φ ) , and Φ is a Φ -ultrametric on X . The next corollary follows from Corollary 3.11 and Corollary 3.19.
Corollary 3.20.
Let ( Q, Q ) be a poset with a smallest element, let X be a nonempty set and let d : X → Q be an ultrametric distance inthe sense of Priess-Crampe and Ribenboim. If the inequality | Q | ℵ holds, then the following conditions are equivalent. ( i ) The mapping d is a Q -ultrametric. ( ii ) There is an usual ultrametric ρ : X → R + such that d and ρ are combinatorially similar. The following proposition guarantees, for a given Q -pseudoultra-metric d , the presence of the weakest (on Q ) partial order at which d remains Q -pseudoultrametric. Proposition 3.21.
Let X be a nonempty set, ( Q, Q ) be a poset andlet d : X → Q be a Q -pseudoultrametric. Then there is a unique partial order Q on Q such that d is a Q -pseudoultrametric and theinclusion Q ⊆ holds whenever is a partial order on Q for which d is a -pseudo-ultrametric.Proof. The uniqueness of Q satisfying the desirable conditions is clear.For the proof of existence of Q , let F = { i : i ∈ I } be the family ofall partial orders i on Q for which d is a i -pseudoultrametric. Thefamily F is non-void because Q ∈ F . Let us define a binary relation Q as the intersection of all i , i.e., for p , q ∈ Q , ( h p, q i ∈ Q ) ⇔ ( p i q holds for every i ∈ I ) . Then Q is a partial order on Q . Since d is a Q -pseudoultrametric,the poset ( Q, Q ) has a smallest element q by definition. It is easyto prove that q is the common smallest element of all posets ( Q, i ) , i ∈ I .Indeed, since d is a Q -pseudoultrametric, we have d ( x, x ) = q . Inaddition, since, for arbitrary i ∗ ∈ I , the mapping d is a i ∗ -pseudo-ultrametric, we also have d ( x, x ) = q ∗ , where q ∗ is the smallest element of ( Q, i ∗ ) . That implies q ∗ = q .Consequently, q is the smallest element of ( Q, Q ) .Hence, to prove that d is a Q -pseudoultrametric it suffices to showthat for every triple h x , x , x i of points of X there is a permutation (cid:18) x x x x i x i x i (cid:19) such that(3.21) d ( x i , x i ) Q d ( x i , x i ) and d ( x i , x i ) = d ( x i , x i ) . Condition (3.21) evidently holds if(3.22) d ( x , x ) = d ( x , x ) = d ( x , x ) . If (3.22) does not hold, then we may set, for definiteness, that(3.23) d ( x , x ) = d ( x , x ) = d ( x , x ) . (The case when d ( x , x ) , d ( x , x ) and d ( x , x ) are pairwise distinctis impossible because d is a Q -pseudoultrametric.) Using (3.23) and(3.9) we obtain(3.24) d ( x , x ) i d ( x , x ) and d ( x , x ) = d ( x , x ) OMBINATORIAL PROPERTIES 23 for every i ∈ I , that, together with the equality Q = \ i ∈ I i , implies d ( x , x ) Q d ( x , x ) and d ( x , x ) = d ( x , x ) . (cid:3) Lemma 3.22.
Let X be a nonempty set, ( Q, Q ) be a poset and let d : X → Q be a Q -pseudoultrametric with d ( X ) = Q . Then theequality (3.25) Q = ( u td ∪ ∆ Q ) holds, where ∆ Q := {h q, q i : q ∈ Q } .Proof. As in the second part of the proof of Theorem 3.10 we see that u td ∪ ∆ Q is reflexive and transitive. Using Q instead of and arguingas in the first part of that proof we obtain the antisymmetry of u td ∪ ∆ Q .Consequently, u td ∪ ∆ Q is a partial order on Q .Let be an arbitrary partial order on Q for which d is a -pseudo-ultrametric. Then, using Definition 3.14 and the definition of u d , wesee that u d ⊆ . The last inclusion implies ( u td ∪ ∆ Q ) ⊆ ( t ∪ ∆ Q ) = . Consequently, Q ⊇ ( u td ∪ ∆ Q ) holds.From the definition of the relation u d , Definition 3.14 and the factthat d is Q -pseudoultrametric it follows that d is a ( u td ∪ ∆ Q ) -pseudo-ultrametric. Thus, equality (3.25) holds. (cid:3) Remark . Equality (3.25) does not hold if d ( X ) = Q . Indeed, if q ∈ Q \ d ( X ) , then we evidently have q / ∈ u td , that implies h q, q i / ∈ ( u td ∪ ∆ Q ) for every q ∈ Q \ { q } . Consequently, the poset ( Q, u td ∪ ∆ Q ) doesnot have any smallest element. The last statement contradicts (3.25),because the smallest element q ∈ d ( X ) of ( Q, Q ) is also the smallestelement of ( Q, Q ) .Results of the present section are based on the fact that, for all posets ( Q, Q ) and ( L, L ) with the smallest elements q ∈ Q and l ∈ L , forevery isotone injection f : Q → L satisfying the condition f ( q ) = l ,and for each Q -pseudoultrametric d , the mappings d and f ◦ d arecombinatorially similar. Moreover, in this case the transformation d f ◦ d converts the Q -pseudoultrametrics into L -pseudoultrametrics. Proposition 3.24.
Let ( Q, Q ) and ( L, L ) be posets with the smallestelements q ∈ Q and l ∈ L . The following conditions are equivalentfor every mapping f : Q → L . ( i ) f ◦ d is a L -pseudoultrametric whenever d is a Q -pseudoultra-metric. ( ii ) f ◦ d is a L -pseudoultrametric whenever d is a Q -ultrametric. ( iii ) f is isotone and f ( q ) = l holds.Proof. ( i ) ⇒ ( ii ) . This is evidently valid. ( ii ) ⇒ ( iii ) . Suppose statement ( ii ) is valid. Then, for every Q -ultrametric space ( X, d ) and for every x ∈ X , the equalities f ( q ) = f ( d ( x, x )) = l hold. Let q , q ∈ Q such that q Q q . We must prove the inequality(3.26) f ( q ) L f ( q ) . This is trivial if f ( q ) = f ( q ) . Suppose f ( q ) = f ( q ) and X = { x , x , x } . Let us define d : X → L as(3.27) d ( x , x ) = d ( x , x ) = q , and d ( x , x ) = q , and d ( x , x ) = d ( x , x ) = d ( x , x ) = q . Then d is a Q -ultrametricand f ◦ d is a L -pseudoultrametric. Inequality (3.26) follows from f ( q ) = f ( q ) , (3.27) and (3.9). ( iii ) ⇒ ( i ) . The validity of this implication follows directly fromthe definition of isotone mappings and the definition of poset-valuedpseudoultrametrics. (cid:3) Corollary 3.25.
Let ( Q, Q ) and ( L, L ) be posets with the small-est elements q ∈ Q and l ∈ L . Then the following conditions areequivalent for every mapping f : Q → L . ( i ) f ◦ d is a L -ultrametric whenever d is a Q -ultrametric. ( ii ) f is isotone and the equivalence (3.28) ( f ( q ) = l ) ⇔ ( q = q ) is valid for every q ∈ Q .Proof. ( i ) ⇒ ( ii ) . Let ( i ) hold. Then, by Proposition 3.24, f is isotoneand f ( q ) = l holds. Thus, to prove ( ii ) it suffices to show that f ( q ) = l implies q = q . Suppose contrary that there is q ∈ Q suchthat q = q and f ( q ) = l .Let X be an arbitrary set with | X | > . The function d : X → Q ,defined as(3.29) d ( x, y ) = ( q if x = y,q if x = y, OMBINATORIAL PROPERTIES 25 is a Q -ultrametric on X . The equalities f ( q ) = l , f ( q ) = l and(3.29) imply f ( d ( x, y )) = l for all x , y ∈ X . Hence, f ◦ d is not a L -ultrametric on X , which contradicts condition ( i ) . ( ii ) ⇒ ( i ) . Suppose ( ii ) holds, but there are a set X and a Q -ultra-metric d : X → Q such that f ◦ d is not a L -ultrametric. Then weevidently have | X | > . Moreover, Proposition 3.24 implies that f ◦ d isa L -pseudoultrametric. Consequently, there are x , x ∈ X such that x = x and(3.30) f ( d ( x , x )) = l . Since d is a Q -ultrametric,(3.31) d ( x , x ) = q holds. From (3.30) and (3.31) it follows that (3.28) is false with q = d ( x , x ) , contrary to condition ( ii ) . (cid:3) The following example shows that we cannot replace statement ( i ) of Corollary 3.25 by the statement • f ◦ d is an ultrametric distance w.r.t ( L, L ) whenever d is anultrametric distance w.r.t ( Q, Q ) leaving statement ( ii ) unchanged. Example 3.26.
Let P and Q be sets with | P | = | Q | > and let P bea linear order on P with a smallest element p . Let us define a binaryrelation Q on Q by the rule:(3.32) ( h q , q i ∈ Q ) ⇔ ( q = q or q = q ) . Then Q is a partial order on Q and, for a set X = { x , x , x } , amapping d : X → Q is an ultrametric distance w.r.t. ( Q, Q ) if andonly if d is symmetric and ( d ( x, y ) = q ) ⇔ ( x = y ) holds for all x , y ∈ X . Since | Q | > holds, there is an ultrametricdistance d ∗ : X → Q such that d ∗ ( x , x ) , d ∗ ( x , x ) , d ∗ ( x , x ) arepairwise distinct. It follows directly from (3.32) and Definition 3.5 thata function f : Q → P is isotone if and only if f ( q ) = p . Now, usingthe equality | P | = | Q | we can find an isotone bijection f ∗ : Q → P suchthat ( f ∗ ( q ) = p ) ⇔ ( q = q ) is valid for every q ∈ Q . Since ( P, P ) is totally ordered, and f ∗ isbijective, and d ∗ ( x , x ) , d ∗ ( x , x ) , d ∗ ( x , x ) are pairwise distinct, we can find a permutation (cid:18) x x x x i x i x i (cid:19) for which f ∗ ( d ∗ ( x i , x i )) ≺ P f ∗ ( d ∗ ( x i , x i )) ≺ P f ∗ ( d ∗ ( x i , x i )) . From Definition 1.3 it follows that the mapping X d ∗ −→ Q f ∗ −→ P is not an ultrametric distance w.r.t. ( P, P ) . Remark . For the case of standard ultrametrics and pseudoultra-metrics Proposition 3.24 and Corollary 3.25 are known. In particular,Proposition 3.24 is a generalization of Proposition 2.4 [18] and, respec-tively, Corollary 3.25 is a generalization of Theorem 9 [55].4.
From weak similarities to combinatorial similaritiesand back
Let us expand the notion of weak similarity to the case of poset-valued pseudoultrametrics.
Definition 4.1.
Let ( Q i , Q i ) be a poset, and ( X i , d i ) be a Q i -pseudo-ultrametric space, and let Y i := d i ( X i ) , i = 1 , . A bijection Φ : X → X is a weak similarity for d and d if there is an isomorphism f : Y → Y of the subposet ( Y , Y ) of the poset ( Q , Q ) and the subposet ( Y , Y ) of the poset ( Q , Q ) such that(4.1) d ( x, y ) = f ( d (Φ( x ) , Φ( y ))) for all x , y ∈ X . Remark . For every totally ordered set ( P , P ) and arbitrary poset ( P , P ) , every isotone bijection f : P → P is an isomorphism of ( P , P ) and ( P , P ) . Thus, Definition 2.12 and Definition 4.1 areequivalent for the case when ( Q , Q ) and ( Q , Q ) coincide with ( R + , ) .The following is a generalization of Proposition 2.14. Proposition 4.3.
Let ( Q i , Q i ) be a poset and ( X i , d i ) be a Q i -pseudoultrametric space, i = 1 , . Then every weak similarity for d and d is a combinatorial similarity for d and d . OMBINATORIAL PROPERTIES 27
Proof.
The proposition can be directly driven from definitions. We justnotice that if Y := d ( X ) and Y := d ( X ) , and f : Y → Y is anisomorphism of the subposet ( Y , Y ) of ( Q , Q ) and the subposet ( Y , Y ) of ( Q , Q ) , and (4.1) holds for all x , y ∈ X , then we have q = f ( q ) , where q i ∈ d i ( X i ) is the smallest element of ( Q i , Q i ) , i = 1 , , that agrees with Proposition 3.16 and the second statementof Proposition 2.4. (cid:3) Theorem 4.4.
Let X i be a nonempty set and let Φ i be a mapping with dom Φ = X i , i = 1 , . Suppose (4.2) := u t Φ ∪ ∆ Φ ( X ) and := u t Φ ∪ ∆ Φ ( X ) are partial orders on Φ ( X ) and, respectively, on Φ ( X ) . If Φ i isa i -pseudoultrametric, i = 1 , , then the following conditions areequivalent for every mapping g : X → X . ( i ) g is a weak similarity for Φ and Φ . ( ii ) g is a combinatorial similarity for Φ and Φ .Proof. Suppose Φ i is a i -pseudoultrametric, i = 1 , . ( i ) ⇒ ( ii ) . This is valid by Proposition 4.3. ( ii ) ⇒ ( i ) . Let g : X → X be a combinatorial similarity. Wemust prove that g is a weak similarity for Φ and Φ . Since g is acombinatorial similarity, there is a bijection f : Φ ( X ) → Φ ( X ) suchthat(4.3) Φ ( x, y ) = f (Φ ( g ( x ) , g ( y ))) holds for all x , y ∈ X . In the correspondence with Definition 4.1, itsuffices to show that f is an isomorphism of the posets (Φ ( X ) , ) and (Φ ( X ) , ) . Using (4.2) we see that if(4.4) (cid:0) h a, b i ∈ u Φ (cid:1) ⇔ (cid:0) h f ( a ) , f ( b ) i ∈ u Φ (cid:1) is valid for all a , b ∈ Φ ( X ) , then f is an isomorphism of these posets.Condition (4.4) follows directly from (4.3) and the definitions of u Φ and u Φ . (cid:3) Corollary 4.5.
Let X and Y be nonempty sets and let ( Q, Q ) and ( L, L ) be posets. Suppose d Q : X → Q and d L : Y → L are a Q -pseudoultrametric and a L -pseudoultrametric, respectively. If d Q ( X ) = Q , and d L ( Y ) = L , and Q = Q , and L = L , then thefollowing conditions are equivalent for every mapping Φ : X → Y . ( i ) Φ is a weak similarity for d Q and d L . ( ii ) Φ is a combinatorial similarity for d Q and d L . In what follows we will use the next modification of Corollary 4.5.
Lemma 4.6.
Let ( Q, Q ) be a totally ordered set and let d : Q → Q be a Q -pseudoultrametric such that d ( Q ) = Q and Q = Q . Then,for every poset ( L, L ) having a smallest element and for each L -pseudoultrametric d L : X → L with d L ( X ) = L , the following state-ment holds. If d L is combinatorially similar to d , then the correspondingcombinatorial similarity is a weak similarity for d and d L .Proof. Let ( L, L ) be a poset with a smallest element and let d L be apseudoultrametric on a set X with d L ( X ) = L . Suppose d and d L arecombinatorially similar. Then there are bijections g : X → Q and f : Q → L such that the diagram(4.5) Q X Q Lg ⊗ gfd d L is commutative. If f is an isomorphism of ( Q, Q ) and ( L, L ) , then g is a weak similarity. Since ( Q, Q ) is totally ordered and f is bi-jective, to prove that f is an isomorphism it suffices to show that theimplication(4.6) ( q Q q ) ⇒ ( f ( q ) L f ( q )) is valid for all q , q ∈ Q . The inclusion L ⊆ L (see Proposition 3.21)implies that (4.6) is valid if(4.7) ( q Q q ) ⇒ ( f ( q ) L f ( q )) . By Lemma 3.22, the equalities d ( Q ) = Q and d ( X ) = L imply(4.8) Q = u td ∪ ∆ Q and L = u td L ∪ ∆ L . Using (4.8) we see that (4.7) is valid whenever ( h q , q i ∈ u d ) ⇒ ( h f ( q ) , f ( q ) i ∈ u d L ) , which follows directly from the commutativity of (4.5) and the defini-tion of u d and u d L . (cid:3) Proposition 4.7.
Let ( Q, Q ) be a totally ordered set with a smallestelement q . Then there is a Q -ultrametric d : Q → Q such that d ( Q ) = Q and Q = Q . OMBINATORIAL PROPERTIES 29
Proof.
Let us define a mapping d : Q → Q by the rule:(4.9) d ( p, q ) := q if p = q,p if q ≺ Q p,q if p ≺ Q q. It is clear that d is symmetric and the equality d ( p, q ) = q holds if andonly if p = q .Now let h q , q , q i be a triple of points of Q . Suppose these pointsare pairwise different. Since ( Q, Q ) is totally ordered, there is a per-mutation (cid:18) q q q q i q i q i (cid:19) such that(4.10) q i ≺ Q q i ≺ Q q i . From (4.9) and (4.10) it follows that d ( q i , q i ) = q i ≺ Q q i = d ( q i , q i ) = d ( q i , q i ) . Thus,(4.11) d ( q i , q i ) d ( q i , q i ) = d ( q i , q i ) holds. Analogously, if the number of different points in h q , q , q i istwo, we can find a permutation such that q i = q i = q i . Hence, d ( q i , q i ) = q ≺ Q d ( q i , q i ) = d ( q i , q i ) , that implies (4.11). For the case when q = q = q holds, (4.11) istrivially valid for every permutation (cid:18) q q q q i q i q i (cid:19) . Hence, d is a Q -ultrametric on Q .It follows from (4.9) that d ( q , q ) = q holds for every q ∈ Q . Thus,we have(4.12) d ( Q ) = Q. To complete the proof it suffices to show that(4.13) Q = Q . By definition of Q , equality (4.13) holds if(4.14) Q ⊇ Q . Lemma 3.22 and (4.12) imply the equality Q = ( u td ∪ ∆ Q ) . Conse-quently, (4.14) is valid if and only if(4.15) ( u td ∪ ∆ Q ) ⊇ Q . Let q and q be some points of Q and let q Q q . If there is q ∈ Q such that(4.16) q = d ( q , q ) and q = d ( q , q ) = d ( q , q ) , then h q , q i ∈ u d holds. If we set q equals to q , the smallest elementof ( Q, Q ) , then (4.16) follows from q Q q and (4.9). Thus, theinclusion u d ⊇ Q holds, that implies (4.15). (cid:3) Remark . If Q is finite, Q = { , , . . . , n } , and Q = hold, thenthe mapping d defined by (4.9) is an ultrametric on Q for which theultrametric space ( Q, d ) is “as rigid as possible”. Some extremal proper-ties of such spaces and related graph-theoretical characterizations werefound in [28]. Example 4.9.
Let us denote by R the Cartesian product of R + andthe two-points set { , } , R := R + × { , } , and let R be the lexico-graphical order on R ,(4.17) (cid:0) h a, b i R h c, d i (cid:1) ⇔ (cid:0) ( a < c ) or ( a = c and b = 0 and d = 1) (cid:1) , where is the standard order on R + . The poset ( R , R ) is totallyordered. By Proposition 4.7, the mapping d : R → R , defined byformula (4.9), is a Q -ultrametric and(4.18) d ( R ) = R and R = R hold.Suppose that there is an ultrametric space ( X, ρ ) such that d and ρ are combinatorially similar. From the definition of combinatorialsimilarity it follows that there are bijections f : ρ ( X ) → d ( R ) and g : R → X such that d ( x, y ) = f ( ρ ( g ( x ) , g ( y ))) holds for all x , y ∈ R .Let us consider now the poset ( ρ ( X ) , ρ ) , where(4.19) ρ := u tρ ∪ ∆ ρ ( X ) . By Theorem 3.18, ρ is a ρ -ultrametric on X . Moreover, usingLemma 3.22 and Theorem 4.4 we obtain that g : R → X is a weaksimilarity for d and ρ . Hence, f : ρ ( X ) → R is an isomorphism of ( R , R ) and ( ρ ( X ) , ρ ) . Proposition 3.21, Lemma 3.22 and (4.19)imply(4.20) ( q ≺ R q ) ⇔ ( f − ( q ) < f − ( q )) for all q , q ∈ R . OMBINATORIAL PROPERTIES 31
Let us consider now the points q xi := h x, i i and q yi := h y, i i , i = 0 , , x, y ∈ R + . It follows directly from (4.17) that if x < y , then q x ≺ R q x ≺ R q y ≺ R q y . Consequently,(4.21) f − ( q x ) < f − ( q x ) < f − ( q y ) < f − ( q y ) . Since Q + = R + ∩ Q is a dense subset of R + , for every x ∈ R + there is p x ∈ Q + such that(4.22) f − ( q x ) < p x < f − ( q x ) . From (4.21) and (4.22) it follows that the mapping R + ∋ x p x ∈ Q + is injective, contrary to the equalities | R + | = 2 ℵ and | Q + | = ℵ . Thus,there are no ultrametrics which are combinatorially similar to d . Remark . An interesting topological property of the poset ( R , R ) was found by F. S.Cater [9]. We will return to it later in Theorem 4.20.Example 4.9 shows that, after replacing ℵ by ℵ and Q + by R + ,Theorem 3.10 becomes false. In particular, we have the followingproposition. Proposition 4.11.
Let X be a set with | X | = 2 ℵ . Then there is ametric d ∗ : X → R + such that: ( i ) If ρ is an arbitrary ultrametric, then ρ and d ∗ are not combi-natorially similar; ( ii ) For every X ⊆ X with | X | ℵ , the restriction d ∗ | X of d ∗ is combinatorially similar to an ultrametric.Proof. Let d : R → R be the R -ultrametric defined in Example 4.9.The equalities(4.23) | X | = 2 ℵ and ℵ = | R | imply the existence of a bijection g : X → R . Let d : X → R be a R -ultrametric defined as d ( x, y ) = d ( g ( x ) , g ( y )) , x, y ∈ X. From (4.23) it follows that | d ( X ) | ℵ . Consequently, by statement ( i ) of Corollary 3.17, there is an usual metric d such that d and d are combinatorially similar. It follows directly from the definition ofcombinatorial similarity that there is a metric d ∗ : X → R + which is combinatorially similar to d . Thus, d ∗ and d are combinatoriallysimilar.It is easy to prove that d ∗ satisfies conditions ( i ) and ( ii ) . Indeed,condition ( ii ) follows from statement ( ii ) of Corollary 3.17. Further-more, it was shown in Example 4.9 that there are no ultrametrics whichare combinatorially similar to d : R → R . Consequently, ( i ) alsoholds. (cid:3) Let ( Q, Q ) be a totally ordered set, and let A , B be nonemptysubsets of Q . We write A ≺ Q B when a ≺ Q b holds for all a ∈ A and b ∈ B .The sets A and B are neighboring if A ≺ Q B or, respectively, B ≺ Q A and there is no q ∈ Q such that A ≺ Q { q } and { q } ≺ Q B or, respectively, B ≺ Q { q } and { q } ≺ Q A. Definition 4.12.
A totally ordered set Q is a η -set if it has no neigh-boring subsets which both have a cardinality strictly less than ℵ .Let ( Q, Q ) and ( L, L ) be posets. An injection f : Q → L is an embedding of ( Q, Q ) in ( L, L ) if (cid:0) q Q q (cid:1) ⇔ (cid:0) f ( q ) L f ( q ) (cid:1) is valid for all q , q ∈ Q .A totally ordered set L is ℵ - universal if every totally ordered set Q with | Q | ℵ can be embedded into L . Lemma 4.13.
Every η -set is ℵ -universal. For the detailed proof of the lemma see, for example, Theorem 20in [1].
Remark . The above definition of ℵ -universal sets can be naturallyextended to arbitrary infinite cardinal number ℵ . The constructionof ℵ -universal posets was studied by many mathematicians (see, forexample, [34, 40] and the references therein).In the proof of the following theorem we will use the ContinuumHypothesis. Theorem 4.15.
Let X be a nonempty set, let Φ be a mapping with dom Φ = X and | Φ( X ) | ℵ , and let ( Q, Q ) be a η -set with asmallest element q . Then the following conditions are equivalent. ( i ) Φ is combinatorially similar to a Q -pseudoultrametric. OMBINATORIAL PROPERTIES 33 ( ii ) The mapping Φ is symmetric, and the transitive closure u t Φ ofthe binary relation u Φ is antisymmetric, and Φ is a -coherentfor a point a ∈ Φ( X ) , and, for every triple h x , x , x i ofpoints of X , there is a permutation (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) .Proof. The validity of ( i ) ⇒ ( ii ) follows from Theorem 3.18.Suppose that ( ii ) holds. Using Theorem 3.18 we obtain that Φ is a Φ -pseudoultrametric for the partial order Φ := u t Φ ∪ ∆ Φ( X ) defined on Φ( X ) .By Lemma 3.7 (Szpilrajn), there is an linear order on Φ( X ) suchthat Φ ⊆ . Consequently, Φ is also a -pseudoultrametric. Theinequality | Φ( X ) | ℵ holds. The Continuum Hypothesis, ℵ = ℵ , and the last inequality imply the inequality | Φ( X ) | ℵ . ByLemma 4.13, the η -set ( Q, Q ) is ℵ -universal. It is easy to prove thatthere is an embedding f : Φ( X ) → Q of (Φ( X ) , ) in ( Q, Q ) suchthat f ( a ) = q . Then the mapping X −→ Φ( X ) f −→ Q is a Q -pseudoultrametric and this mapping is combinatorially similarto Φ . (cid:3) The following definition can be found in [41, pp. 57–58].
Definition 4.16.
Let ( Q, Q ) be a totally ordered set with | Q | > .A topology τ with a subbase consisting of all sets of the form { q ∈ Q : q ≺ Q a } or { q ∈ Q : a ≺ Q q } for some a ∈ Q is the order topology on Q . In this case we say that τ is the Q -topology for short.Recall that a topological space is second countable if it has a count-able or finite base. Lemma 4.17.
Let ( Q, Q ) be a totally ordered set with | Q | > . Thenthe following conditions are equivalent. ( i ) The Q -topology is second countable. ( ii ) The poset ( Q, Q ) is isomorphic to a subposet of ( R + , ) . This lemma is a simple modification of Theorem II from paper [9] ofF. S. Cater.
Theorem 4.18.
Let ( Q, Q ) be a totally ordered set satisfying | Q | > and having the smallest element q . Then the following conditions areequivalent. ( i ) The Q -topology is second countable. ( ii ) For every Q -pseudoultrametric d there is a pseudoultrametric ρ such that d and ρ are weakly similar. ( iii ) For every Q -pseudoultrametric d there is a pseudoultrametric ρ such that d and ρ are combinatorially similar.Proof. It is easy to see that ( i ) , ( ii ) and ( iii ) are equivalent if | Q | = 2 .Suppose | Q | > holds. ( i ) ⇒ ( ii ) . Let the Q -topology be second countable, let X be anonempty set and let d : X → Q be a Q -pseudoultrametric. Write Q := Q \ { q } and Q := Q ∩ Q . The inequality | Q | > implies | Q | > . The Q -topology coincides with the topology induced on Q by Q -topology. Consequently, the Q -topology is also secondcountable. Hence, by Lemma 4.17, there is an isomorphism f : Q → A of the posets ( Q , Q ) and ( A , ) , where A ⊆ (0 , ∞ ) and is thestandard order on R . Write A := A ∪ { } . The function f ∗ : Q → A , f ∗ ( q ) = ( if q = q ,f ( q ) if q = q , is an isomorphism of ( Q, Q ) and ( A, ) . Let ρ : X → R + be definedas ρ ( x, y ) = f ∗ ( d ( x, y )) , x, y ∈ X. Then ρ is a pseudoultrametric on X and the identical mapping X id −→ X is a weak similarity for d and ρ . ( ii ) ⇒ ( iii ) . The validity of this implication follows from Proposi-tion 4.3. ( iii ) ⇒ ( i ) . Suppose condition ( iii ) holds. By Proposition 4.7, thereis a Q -ultrametric d : Q → Q satisfying the equalities d ( Q ) = Q and Q = Q .Let ρ : X → R + be a pseudoultrametric such that ρ and d arecombinatorially similar. Write L := ρ ( X ) and L := ∩ L . Thenthe L -pseudoultrametric ρ L : X → L , ρ L ( x, y ) = ρ ( x, y ) , x, y ∈ X, is also combinatorially similar to d . By Lemma 4.6, d and ρ L are weaklysimilar. Using Definition 4.1 we obtain that ( Q, Q ) is isomorphic tothe subposet ( L, L ) of ( R + , ) . Hence, by Lemma 4.17 (Cater), the Q -topology is second countable. (cid:3) OMBINATORIAL PROPERTIES 35
Recall that a topological space ( X, τ ) is said to be separable if thereis a set A ⊆ X such that | A | ℵ and A ∩ U = ∅ for every nonemptyset U ∈ τ .In what follows we denote by ( R , R ) the totally ordered set con-structed in Example 4.9.The next lemma is a part of Theorem III [9]. Lemma 4.19 (Cater) . Let ( Q, Q ) be a totally ordered set with | Q | > . Then the following conditions are equivalent. ( i ) The Q -topology is separable. ( ii ) The poset ( Q, Q ) is isomorphic to a subposet of ( R , R ) . Theorem 4.20.
Let ( Q, Q ) be a totally ordered set having a small-est element and satisfying the inequality | Q | > . Then the followingconditions are equivalent. ( i ) The Q -topology is separable. ( ii ) For every Q -pseudoultrametric d there is a R -pseudoultra-metric ρ such that d and ρ are weakly similar. ( iii ) For every Q -pseudoultrametric d there is a R -pseudoultra-metric ρ such that d and ρ are combinatorially similar. Using Lemma 4.19 instead of Lemma 4.17 we can prove this theoremsimilarly to Theorem 4.18.The following theorem gives us some necessary and sufficient condi-tions under which a mapping is combinatorially similar to a pseudoul-trametric, and it can be considered as a main result of the section.
Theorem 4.21.
Let X be a nonempty set and let Φ be a mapping with dom Φ = X . Then the following conditions are equivalent. ( i ) Φ is combinatorially similar to pseudoultrametric. ( ii ) There is b ∈ Φ( X ) such that Φ( x, x ) = b holds for every x ∈ X , and the binary relation (4.24) Φ := u t Φ ∪ ∆ Φ( X ) is a partial order on Φ( X ) , and b is the smallest elementof (Φ( X ) , Φ ) , and Φ is a Φ -pseudoultrametric on X , andthere is a linear order on Φ( X ) such that (4.25) Φ ⊆ holds, and (Φ( X ) , ) is isomorphic to a subposet of ( R + , ) . ( iii ) The mapping Φ is symmetric, and there is a ∈ Φ( X ) forwhich Φ is a -coherent, and, for every triple h x , x , x i of points of X , there is a permutation (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) , and there is a linear order on Φ( X ) such that a is the smallest element of (Φ( X ) , ) and u Φ ⊆ holds, and (Φ( X ) , ) is isomorphic to a subposetof ( R + , ) .Proof. ( i ) ⇒ ( ii ) . Let ( i ) hold. Then using Theorem 3.18 we see thatcondition ( ii ) is valid whenever there is a linear order on Φ( X ) such that (4.25) holds and (Φ( X ) , ) is isomorphic to a subposet of ( R + , ) .By condition ( i ) , there are a set Y and a pseudoultrametric ρ : Y → R + such that Φ and ρ are combinatorially similar. Write(4.26) ρ := u tρ ∪ ∆ ρ ( Y ) . From Lemma 3.22 it follows that ρ is a ρ -pseudoultrametric. Since Φ and ρ are combinatorially similar, there exists a bijection g : X → Y such that g is combinatorial similarity for Φ and ρ . Now usingTheorem 4.4, and (4.24), and (4.26) we see that g is a weak similarityfor Φ and ρ . Consequently, there is an order isomorphism f : Φ( X ) → ρ ( Y ) of posets (Φ( X ) , Φ ) and ( ρ ( Y ) , ρ ) . By Proposition 3.21 andLemma 3.22, we obtain that ( γ ρ γ ) ⇒ ( γ γ ) is valid for all γ , γ ∈ ρ ( Y ) .Let us define a binary relation by the rule: ( h g , g i ∈ ) ⇔ ( h g , g i ∈ Φ( X ) × Φ( X ) and ( f ( g ) f ( g ))) Then is a linear order satisfying all desirable conditions. ( ii ) ⇒ ( i ) . Suppose ( ii ) holds. Then Φ is a Φ -pseudoultrametricon X and there is an injection f : Φ( X ) → R + such that ( b Φ b ) ⇒ ( f ( b ) f ( b )) holds for all b , b ∈ Φ( X ) . Since b is the smallest element of theposet (Φ( X ) , Φ ) , the function f ∗ : Φ( X ) → R + defined as f ∗ ( b ) = f ( b ) − f ( b ) is nonnegative and isotone, and satisfies the condition ( f ∗ ( b ) = 0) ⇔ ( b = b ) OMBINATORIAL PROPERTIES 37 for every b ∈ Φ( X ) . Proposition 3.24 implies that f ∗ ◦ Φ is a pseu-doultrametric on X . From Definition 1.4 it directly follows that Φ and f ∗ ◦ Φ are combinatorially similar.The validity of the equivalence ( ii ) ⇔ ( iii ) follows from Theo-rem 3.18. We only note that u t Φ is antisymmetric if and only if thereis a partial order ′ such that ′ ⊇ u Φ . (cid:3) The proof of the following corollary is similar to prove of Theo-rem 4.21.
Corollary 4.22.
Let X be a nonempty set and let Φ be a mapping with dom Φ = X . Then the following conditions are equivalent. ( i ) Φ is combinatorially similar to ultrametric. ( ii ) There is b ∈ Φ( X ) such that Φ − ( b ) = ∆ X , and the binaryrelation Φ := u t Φ ∪ ∆ Φ( X ) is a partial order on Φ( X ) , and b is the smallest element of (Φ( X ) , Φ ) , and Φ is a Φ -ultrametric on X , and there is alinear order on Φ( X ) such that Φ ⊆ holds, and (Φ( X ) , ) is isomorphic to a subposet of ( R + , ) . ( iii ) The mapping Φ is symmetric, and there is a ∈ Φ( X ) forwhich Φ − ( a ) = ∆ X holds, and, for every triple h x , x , x i of points of X , there is a permutation (cid:18) x x x x i x i x i (cid:19) such that Φ( x i , x i ) = Φ( x i , x i ) , and there is a linear order on Φ( X ) such that a is the smallest element of (Φ( X ) , ) and u Φ ⊆ holds, and (Φ( X ) , ) is isomorphic to a subposetof ( R + , ) . In connection with Theorem 4.21 and Corollary 4.22, the followingproblem naturally arises.
Problem 4.23.
Describe (up to order-isomorphism) the partially or-dered sets ( Q, Q ) which admit extensions to totally ordered sets ( Q, ) such that ( Q, ) is order-isomorphic to a subposet of ( R + , ) .We do not discuss this problem in details but formulate the followingconjecture. Conjecture 4.24.
The following conditions are equivalent. ( i ) A poset ( Q, Q ) admits an extension to totally ordered set ( Q, ) such that ( Q, ) is order-isomorphic to a subposet of ( R + , ) . ( ii ) The inequality | Q | ℵ holds and every totally ordered sub-poset of ( Q, Q ) can be embedded into ( R + , ) . References
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O. Dovgoshey
Function theory departmentInstitute of Applied Mathematics and Mechanics of NASUDobrovolskogo str. 1, Slovyansk 84100, Ukraine
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