Ultradiversification of Diversities

In this paper, using the idea of ultrametrization ofmetric spaceswe introduce ultradiversi cation of diversities. We show that every diversity has an ultradiversi cation which is the greatest nonexpansive ultradiversity image of it. We also investigate a Hausdor -Bayod type problem in the setting of diversities, namely, determining what diversities admit a subdominant ultradiversity. This gives a description of all diversities which can be mapped onto ultradiversities by an injective nonexpansive map. The given results generalize similar results in the setting of metric spaces.


Introduction and Preliminaries
An ultrametric space is a metric space (X, d) in which the distance function d satis es the strong triangle inequality d(x, z) ≤ max{d(x, y), d(y, z)}, for all x, y, z ∈ X. A description of all metric spaces which can be mapped onto ultrametric spaces by an injective nonexpansive map is given in [7]. Indeed, it is shown that for any metric space (X, d) there exists an ultrametrization of X which is the greatest nonexpansive ultrametric image of (X, d). This, in particular, determines that the category of ultrametric spaces and nonexpansive maps is a re ective subcategory in the category of all metric spaces and the nonexpansive maps. Moreover, a complete solution of the Hausdor -Bayod problem, namely, determining what metric spaces admit a subdominant ultrametric is given in [7]. In fact, the Hausdor -Bayod problem for nonexpansive injective maps of metric spaces is that "For what metric spaces (X, d) does there exist an ultrametric ∆ on X such that the identity map i : (X, d) → (X, ∆) is nonexpansive?"ds (see [8] and references therein).
On the other hand, diversities were introduced in [2] as a generalization of metric spaces and tight span of metric spaces was developed by diversities. Recently, some other aspects of metric space theory carried over to diversities (see e.g., [4,6]). In addition, a diversity counterpart of ultrametric spaces was introduced in [9] under the name "Ultradiversity".
In this paper, inspired by the ultrametrization method of metric spaces given in [7], we show that for any diversity (X, δ) there exists an ultradiversi cation of X which is the greatest nonexpansive ultradiversity image of the diversity (X, δ) (Theorem 2.1). In addition, the question that whether for any diversity there exists an ultradiversity smaller than it leads us to investigate a Hausdor -Bayod type problem in the setting of diversities, i.e., determining that what diversities admit a subdominant ultradiversity (Theorem 2.2).
In order to introduce the ultradiversi cation of diversities, an analogous notion to ultrametrization of metric spaces, we need to review some notions. We start with some de nitions and preliminaries regarding diversities and ultrametrization of metric spaces.
De nition 1.1 [9] An ultradiversity is a pair (X, δ) in which X is a nonempty set and δ : X → R is a real function on the set of all nite subsets X of X satisfying: Notice that each ultradiversity (X, δ) is also a diversity, i.e., in addition to (UD1) and (UD2) it satis es the condition: [2]). For recent works on diversities we also refer to [3][4][5][6].
It is worth mentioning that for every ultradiversity (diversity) (X, δ), the function d : X × X → R de ned as d(x, y) = δ({x, y}), for all x, y ∈ X, is an ultrametric (a metric), called the induced ultrametric (metric) for (X, δ). Furthermore, every diversity (and therefore ultradiversity) δ enjoys the monotonicity property, i.e., A ⊆ B implies δ(A) ≤ δ(B). From (UD2) and the monotonicity of the ultradiversity δ, it is easy to see that if (1.1) where A ∈ X . Then (X, δ) is an ultradiversity which is called the induced diameter ultradiversity for the ultrametric space (X, d) (or brie y, for the ultrametric d). Furthermore, it can be seen that every where T is the maximum edge weight along T. Then δ is an ultradiversity on vertices of G (see Figure 1). Indeed, without loss of generality suppose that δ(A ∪ B) ≤ δ(B ∪ C) and let T be a tree containing B ∪ C. Thus S ≤ T , for some tree S containing A ∪ B. There obviously exists a tree R containing A ∪ C consisting of edges of P and T with R = T . Therefore δ(A ∪ C) ≤ δ(B ∪ C) which shows δ satis es (UD2).
and all organisms of A belong to the same species n all organisms of A belong to a same n th taxonomic rank, but not to a same n − th one.
Then δ is an ultradiversity on the set of all organisms.
for all A ∈ X . Then δ is an ultradiversity on X.
The next example is in a more general form than the previous example.
where A ∈ X is an ultradiversity on X.
Now, we review some concepts given in [7]. We recall that a map f : for all x, y ∈ X. Let (X, d) be a metric space. By [7,Theorem 5], there are an ultrametric space (uX, du) and a nonexpansive surjection u : (X, d) → (uX, du) such that for any nonexpansive map f : (X, d) → (Y , r), where (Y , r) is an arbitrary ultrametric space, there exists a unique nonexpansive map uf : (uX, du) → (Y , r) that commutes the following diagram, i.e., uf • u = f : Then, the ultrametric space (uX, du) is called an ultrametrization of the metric space (X, d).
x and y are ε-linkable}, for all x, y ∈ X enjoys the strong triangle inequality, while the property that ∆(x, y) = implies x = y may not be valid generally. Consider the equivalence relation ∼ on X given by "x ∼ y if and only if x and y are ε-linkable, for every ε > ". Let [x] be the equivalence class of a point x, uX be the quotient set X ∼, and u be the canonical projection map. Then the function du de ned as is an ultrametric on uX, and u : (X, d) → (uX, du) is a nonexpansive surjection (since every pair (x, y) is r) is an ultrametric space, then the map uf : (uX, du) → (Y , r) de ned as is a nonexpansive map which is clearly unique with the property that uf • u = f . Thus, every metric space has an ultrametrization.
In the next section, we introduce the ultradiversi cation of diversities. The given results generalize similar results of [7]. [2]). Notice that for any nonexpansive map f :

Ultradiversi cation
is also nonexpansive, where d X and d Y are the metrics induced by δ X and δ Y , respectively. Moreover, two diversities (X, δ X ) and (Y , δ Y ) are said to be isomorphic if there exists a bijective map f : We say that a nite subset A of a diversity (X, δ) is ε-linkable if each two elements a and b of A are ε-linkable with respect to the induced metric of δ (or equivalently, if there exists an ε-tree T containing A, i.e., a tree T = (V , E) on the underlying set X with δ({u, v}) ≤ ε, for every edge {u, v} ∈ E, and A ⊆ V). Moreover (X, δ) is said to be totally unlinked if its induced metric is so, i.e., each two elements x and y of X are not ε-linkable, for some positive number ε (see [7] and [8]). Example 1.1 shows that any ultrametric space induces an ultradiversity, namely, the diameter ultradiversity. Unlike the variety of diversities (see the diversities in [2][3][4][5][6]), ultradiversities have a common intrinsic form. The following result allows us to consider every ultradiversity as a diameter ultradiversity.
Proof. Let (X, d) be the induced metric space of (X, δ). Let (uX, du) be the ultrametrization of (X, d) de ned as (1.3), and u : X → uX be the canonical projection map. If A = {a , a , · · · , an} is a nite subset of X, then du(u(a i ), u(a j )) = max ≤i,j≤n du(u(a i ), u(a j )), for some i and j . Let δu be the diameter diversity of du. Since u is nonexpansive in the sense of metrics and δ is monotone, we have This implies that u : (X, δ) → (uX, δu) is also nonexpansive in the sense of diversities. We call the ultradiversity (uX, δu) given in Theorem 2.1 an ultradiversi cation of the diversity (X, δ). In fact, it can also be considered as the greatest nonexpansive ultradiversity image of (X, δ). To see this, let (X, ∆) be such an ultradiversity with a corresponding surjection nonexpansive map u : (X, δ) → (X, ∆), i.e., for every nonexpansive map f from (X, δ) to an arbitrary ultradiversity (Y , σ) we have . Thus f (x) = f (y) and so g is well-de ned. The nonexpansivity of g can be easily seen from (2.1) and g is clearly the unique map with the property g • u = f . Thus (X, ∆) is an ultradiversi cation of (X, δ). On the other hand, every ultradiversi cation (X, ∆) of (X, δ) has obviously the property (2.1).

Remark 2.1
According to the method given in [7], to reach an ultrametrization of a metric space, an alternative way can also be used to identify the ultradiversi cation diam du (Theorem 2.1). Indeed, de ne To see this, we rst show that for every A ∈ X we have max{∆(a, b) : a, b ∈ A}.

By de nition, we see that if
Conversely, suppose that a and b are two elements of A which maximize ∆(a, b) and ε is a positive real number. Since each two elements a and b in A are ∆(a, b)  Having noted to the balls drawn, one can see that if α is the minimum distance between points of the n th line and the n + th line, which is equal to the distance between the two black points, then there obviously exists an α-tree containing A, while for no ε smaller than this distance A is ε-linkable.
It is clear that every ultradiversity is an ultradiversi cation of itself. In addition, it is not hard to see that any two diversities on a set X with equivalent induced metrics can have the same uX. To illustrate it more, let us see the diversities given in the following example.
Then, each class of the form [(x, x + n )] can be identi ed with n and therefore uX with N. In addition, for every nite subset A = [(x , x + n )], · · · , [(x k , x k + n k )] of uX, without loss of generality we can assume that n < · · · < n k . Then, 1. For any diversity δ E on X which has Euclidean metric as its induced metric, we have 2. For the -diversity δ given by where A ∈ X (see [4]), we have (δ ) u (A) = n (n + ) .
3. For the ∞-diversity δ∞ which is in fact the diameter diversity of the supremum metric d∞, i.e., where A ∈ X , we have (δ∞) u (A) = n (n + ) .
An intuition of the ultradiversi cation δu of the diversities (X, δ) given in Example 2.1 can be seen in Figure 2.

Example 2.2
Let δ be any diversity in R k which induces the dp-metric (the standard metric of the classical space p ) on R k , for some p ∈ [ , ∞]. Since each two elements x and y of R k are ε-linkable for any positive real number ε, the trivial diversity on any singleton can be considered as an ultradiversi cation of (R k , δ).

Example 2.3
Let (X, σ) be a nite diversity and G be the complete graph on vertices X with edge weight σ({x, y}) for every edge {x, y}. Let δ be de ned as in (1.2). Then (X, δ) is an ultradiversi cation for (X, σ).
The previous example and the fact that every ultradiversity is an ultradiversi cation of itself allow us to consider every nite ultradiversity as that of given in Example 1.2.
By the next example it is seen that di erent diversities can have a same ultradiversi cation.

Example 2.4 Let
Let X = D × R and δ be a diversity on X with induced metric dp, where ( ≤ p ≤ ∞). Then y)], for all x ∈ (−∞, ), y ∈ R, and n ∈ N. Note also thatn = {n} × R = [(n, y)], for all y ∈ R and n ∈ N. The canonical projection map u is for all (x, y) ∈ X. Now an ultradiversi cation δu is given as in which m = sup{x : x ∈ R and u (x, ) ∈ A}.
While the following proposition gives a characterization of ultradiversities, it is also a generalization of Lemma 6 in [7].

Proposition 2.2 A diversity (X, δ) is an ultradiversity if and only if no nite subset A of X is ε-linkable for any ε < δ(A). In particular, every ultradiversity is totally unlinked.
Proof. Let (X, δ) be an ultradiversity and A be a nite subset of X. By induction on the cardinal number of A we show that if A is ε-linkable, then δ(A) ≤ ε. This is trivial when |A| = . We also assume that this is true for every n-point subset of X. If A is an ε-linkable subset of X with |A| = n + , then there is an ε-tree T = (V , E) containing A. Let u be an arbitrary leaf of T and v be the vertex for which {u, v} ∈ E. Since δ ({u, v}) ≤ ε and A \ {u} is an n-point ε-linkable subset of X, from equation (1.1) we have Conversely, if (X, δ) is not an ultradiversity, then there exist A, B, C ∈ X such that δ( In fact, if A consists of n points a , · · · , an, then the graph (A, Let (X, d) be a metric space. Given the collection {d i : i ∈ I} of ultrametrics on X dominated by d, the ultrametric is the largest ultrametric on X dominated by d which is called the subdominant ultrametric of the metric d (see [1] in which supremum is taken over all ultradiversities ∆ on X dominated by δ, then (X, ψ) is the induced diameter diversity for the subdominant ultrametric s of d. Furthermore, (X, ψ) is an ultradiversi cation of (X, δ) provided that (X, δ) is totally unlinked.
Proof. Suppose that δ is totally unlinked. If a nite subset A of X is ε-linkable for any ε > , then it does not have more than one point. Therefore, the function ∆ de ned as (2.2) satis es (UD1). Further, since by (2.3) we have diam du (u(A)) = ∆(A), it also satis es (UD2). Thus ∆ is an ultradiversity which is clearly dominated by δ since every A ∈ X is δ(A)-linkable. Conversely, let ∆ ≤ δ for some ultradiversity ∆. By the fact that every ultradiversity is totally unlinked and the fact that for two diversities δ and δ on X such that δ ≤ δ , if δ is totally unlinked, so is δ we imply that δ is totally unlinked. Now, let ∆ be an ultradiversity dominated by δ and d ∆ be the induced metric for ∆. Let A be a nite subset of X, and a and b in A be such that diams(A) = s(a , b ). Since d ∆ ≤ d, we have d ∆ ≤ s and therefore d ∆ (a, b) ≤ s(a, b) ≤ s(a , b ), for all a, b ∈ A. Thus, by Proposition 2.1, ∆(A) = diam d ∆ (A) ≤ diams(A). Hence, ψ ≤ diams. On the other hand, since diams(A) ≤ diam d (A) ≤ δ(A), for any nite subset A of X, we have ψ = diams. Now we show that ∆ = ψ. By the de nition of ψ, it is obvious that ∆ ≤ ψ. For the reverse, suppose that ∆ is an ultradiversity on X dominated by δ. Also, suppose that A is a nite subset of X which is ε-linkable for some ε, and ∆(A) = ∆({a, b}) for some a, b ∈ A. There is a nite sequence (xn) N n= of elements of X with x = a and x N = b such that d(xn , x n+ ) ≤ ε, for all n < N. By (1.1) we have This implies that ψ ≤ ∆. Finally, as (X, δ) is totally unlinked, we have [x] = {x}, for all x ∈ X and therefore two ultradiversities uX, diam du and (X, ∆) are obviously isomorphic (see (2.3)). Thus, (X, ψ) is an ultradiversi cation of (X, δ). Theorem 2.2 describes all diversities (X, δ) which can be mapped onto an ultradiversity (X, ∆) by an injective nonexpansive map f . In particular, if such f : (X, δ) → (X, ∆) exists, then the identity map i : (X, δ) → (X, ∆ ) is nonexpansive in which ∆ is an ultradiversity de ned as ∆ (A) = ∆(f (A)), for all A ∈ X . We also have the following.