Modular Locally Constant Mappings in Vector Ultrametric Spaces

Abstract

We give some sufficient conditions for mappings defined on vector ultrametric spaces to be modular locally constant.

1. Introduction and Preliminaries

A metric space (𝑋,𝑑) in which the triangle inequality is replaced by 𝑑𝑥,𝑦≤max𝑑(𝑥,𝑧),𝑑𝑧,𝑦,𝑥,𝑦,𝑧∈𝑋, (1.1) is called an ultrametric space. Generalized ultrametric spaces were given in [1, 2] via partially ordered sets and some applications of them appeared in logic programming [3], computational logic [4], and quantitative domain theory [5].

In [6], the notion of a metric locally constant function on an ultrametric space was given in order to investigate certain groups of isometries and describe various Galois groups over local fields. Locally constant functions also appear in contexts such as higher ramification groups of finite extensions of 𝐐_𝑝, and the Fontaine ring 𝐵⁺_dR. Also, metric locally constant functions were studied in [7, 8]. On the other hand, vector ultrametric spaces are given in [9] as vectorial generalizations of ultrametrics. Hence, locally constant functions, in modular sense, can play the same role in vector ultrametric spaces as they do in usual ultrametric spaces.

In this paper, we introduce modular locally constant mappings in vector ultrametric spaces. Some sufficient conditions are given for mappings defined on vector ultrametric spaces to be modular locally constant.

We first present some basic notions.

Recall that a modular on a real linear space 𝒜 is a real valued functional 𝜌 on 𝒜 satisfying the conditions: (1)𝜌(𝑥)=0 if and only if 𝑥=0, (2)𝜌(𝑥)=𝜌(−𝑥), (3)𝜌(𝛼𝑥+𝛽𝑦)≤𝜌(𝑥)+𝜌(𝑦), for all 𝑥,𝑦∈𝒜 and 𝛼,𝛽≥0, 𝛼+𝛽=1.

Then, the linear subspace 𝒜_𝜌=𝑥∈𝒜∶𝜌(𝛼𝑥)⟶0as𝛼⟶0 (1.2) of 𝒜 is called a modular space.

A sequence (𝑥_𝑛)^∞_𝑛=1 in 𝒜_𝜌 is called 𝜌-convergent (briefly, convergent) to 𝑥∈𝒜_𝜌 if 𝜌(𝑥_𝑛−𝑥)→0 as 𝑛→∞, and is called Cauchy sequence if 𝜌(𝑥_𝑚−𝑥_𝑛)→0 as 𝑚,𝑛→∞. By a 𝜌-closed (briefly, closed) set in 𝒜_𝜌 we mean a set which contains the limit of each of its convergent sequences. Then, 𝒜_𝜌 is a complete modular space if every Cauchy sequence in 𝒜_𝜌 is convergent to a point of 𝒜_𝜌. We refer to [10, 11] for more details.

A cone 𝒫 in a complete modular space 𝒜_𝜌 is a nonempty set such that

(i) 𝒫 is 𝜌-closed, and 𝒫≠{0};

(ii) 𝑎,𝑏∈ℝ, 𝑎,𝑏≥0, 𝑥,𝑦∈𝒫⇒𝑎𝑥+𝑏𝑦∈𝒫;

(iii) 𝒫∩(−𝒫)={0}, where −𝒫={−𝑥∶𝑥∈𝒫}.

Let ⪯ be the partial order on 𝒜_𝜌 induced by the cone 𝒫, that is, 𝑥⪯𝑦 whenever 𝑦−𝑥∈𝒫. The cone 𝒫 is called normal if 0⪯𝑥⪯𝑦⟹𝜌(𝑥)≤𝜌𝑦,𝑥,𝑦∈𝒜_𝜌. (1.3) The cone 𝒫 is said to be unital if there exists a vector 𝑒∈𝒫 with modular 1 such that 𝑥⪯𝜌(𝑥)𝑒,(𝑥∈𝒫). (1.4)

Example 1.1. Consider the real vector space 𝐶[0,1] consisting of all real-valued continuous functions on [0,1] equipped with the modular 𝜌 defined by 𝜌(𝑥)=max_[]𝑡∈0,1|𝑥(𝑡)|²,(𝑥∈𝐶[0,1]). (1.5) It is not difficult to see that 𝐶[0,1] is a complete modular space and 𝒫={𝑥∈𝐶[0,1]∶𝑥(𝑡)≥0,∀𝑡∈[0,1]} (1.6) is a normal cone in 𝐶[0,1].

Example 1.2. The vector space 𝐶¹[0,1] consisting of all continuously differentiable real-valued functions on [0,1] equipped with the modular 𝜌 defined by 𝜌(𝑥)=max_[]𝑡∈0,1|𝑥(𝑡)|+max_[]𝑡∈0,1𝑥^′(𝑡),𝑥∈𝐶¹[0,1] (1.7) constitutes a complete modular space. The subset 𝒫=𝑥∈𝐶¹[0,1]∶𝑥(𝑡)≥0,∀𝑡∈[0,1] (1.8) is a unital cone in 𝐶¹[0,1] with unit 1. The cone 𝒫 is not normal since, for example, 𝑥(𝑡)=𝑡^𝑛⪯1, for 𝑛≥1 does not imply that 𝜌(𝑥)≤𝜌(1).

Throughout this note, we suppose that 𝒫 is a cone in complete modular space 𝒜_𝜌, and ⪯ is the partial order induced by 𝒫.

Definition 1.3. A vector ultrametric on a nonempty set 𝒳 is a mapping 𝑑∶𝒳×𝒳→𝒜_𝜌 satisfying the conditions:(CUM1)𝑑(𝑥,𝑦)≽0 for all 𝑥,𝑦∈𝒳 and 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦; (CUM2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝒳; (CUM3) If 𝑑(𝑥,𝑧)⪯𝑝 and 𝑑(𝑦,𝑧)⪯𝑝, then 𝑑(𝑥,𝑦)⪯𝑝, for any 𝑥,𝑦,𝑧∈𝒳, and 𝑝∈𝒫. Then the triple (𝒳,𝑑,𝒫) is called a vector ultrametric space. If 𝒫 is unital and normal, then (𝒳,𝑑,𝒫) is called a unital-normal vector ultrametric space.

For unital-normal vector ultrametric space (𝒳,𝑑,𝒫), since 𝑑𝑥,𝑦⪯𝜌𝑑𝑥,𝑦𝑒,𝑑𝑦,𝑧⪯𝜌𝑑𝑦,𝑧𝑒, (1.9) from (CUM3) we get 𝑑(𝑥,𝑧)⪯max𝜌𝑑𝑥,𝑦,𝜌𝑑𝑦,𝑧𝑒, (1.10) and therefore 𝜌(𝑑(𝑥,𝑧))≤max𝜌𝑑𝑥,𝑦,𝜌𝑑𝑦,𝑧. (1.11) Let (𝒳,𝑑,𝒫) be a unital-normal vector ultrametric space. If 𝑥∈𝒳 and 𝑝∈𝒫⧵{0}, the ball 𝐵(𝑥,𝑝) centered at 𝑥 with radius 𝑝 is defined as 𝐵𝑥,𝑝∶=𝑦∈𝒳∶𝜌𝑑𝑥,𝑦≤𝜌𝑝. (1.12) The unital-normal vector ultrametric space (𝒳,𝑑,𝒫) is called spherically complete if every chain of balls (with respect to inclusion) has a nonempty intersection.

The following lemma may be easily obtained.

Lemma 1.4. Let (𝒳,𝑑,𝒫) be a unital-normal vector ultrametric space. (1)If 𝑎,𝑏∈𝒳, 0⪯𝑝 and 𝑏∈𝐵(𝑎,𝑝), then 𝐵(𝑎,𝑝)=𝐵(𝑏,𝑝). (2)If 𝑎,𝑏∈𝒳, 0≺𝑝⪯𝑞, then either 𝐵(𝑎,𝑝)⋂𝐵(𝑏,𝑞)=∅ or 𝐵(𝑎,𝑝)⊆𝐵(𝑏,𝑞).

Definition 1.5. Let (𝒳,𝑑,𝒫) be a unital-normal vector ultrametric space. A mapping 𝑓∶𝒳→𝒫⧵{0} is said to be modular locally constant provided that for any 𝑥∈𝒳 and any 𝑦∈𝐵(𝑥,𝑓(𝑥)) one has 𝜌(𝑓(𝑥))=𝜌(𝑓(𝑦)).

2. Main Theorem

Theorem 2.1. Let (𝒳,𝑑,𝒫) be a spherically complete unital-normal vector ultrametric space and 𝑇∶𝒳→𝒳 be a mapping such that for every 𝑥,𝑦∈𝒳,𝑥≠𝑦, either 𝜌𝑑𝑇𝑥,𝑇𝑦<max𝜌(𝑑(𝑥,𝑇𝑥)),𝜌𝑑𝑦,𝑇𝑦 (2.1) or 𝜌𝑑𝑇𝑥,𝑇𝑦≤𝜌𝑑𝑥,𝑦. (2.2) Then there exists a subset 𝐵 of 𝒳 such that 𝑇∶𝐵→𝐵 and the mapping 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥),(𝑥∈𝐵) (2.3) is modular locally constant.

Proof. Let ℰ={𝐵_𝑎}_𝑎∈𝒳 where 𝐵_𝑎=𝐵(𝑎,𝑑(𝑎,𝑇𝑎)). Consider the partial order ⊑ on ℰ defined by 𝐵_𝑎⊑𝐵_𝑏iff𝐵_𝑏⊆𝐵_𝑎, (2.4) where 𝑎,𝑏∈𝒳. If ℰ₁ is any chain in ℰ, then the spherical completeness of (𝒳,𝑑,𝒫) implies that the intersection Ω of elements of ℰ₁ is nonempty.
Suppose that (2.1) holds. Let 𝑏∈Ω and 𝐵_𝑎∈ℰ₁. Obviously 𝑏∈𝐵_𝑎, so 𝜌(𝑑(𝑎,𝑏))≤𝜌(𝑑(𝑎,𝑇𝑎)). For any 𝑥∈𝐵_𝑏, we have𝜌(𝑑((𝑑(𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑(𝜌(𝑑((𝑑(()𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑((𝑑(,𝜌(𝑑(𝜌(𝑑((𝑑((𝑑(.𝑥,𝑏))≤𝜌𝑏,𝑇𝑏))≤max𝑏,𝑎)),𝜌𝑎,𝑇𝑎)),𝜌𝑇𝑎,𝑇𝑏))<max𝑏,𝑎)),𝜌𝑎,𝑇𝑎)),max𝑎,𝑇𝑎)),𝜌𝑏,𝑇𝑏))by2.1≤max𝑏,𝑎)),𝜌𝑎,𝑇𝑎)),𝜌𝑏,𝑇𝑏))≤max𝑎,𝑇𝑎)),𝜌𝑏,𝑇𝑏))=𝜌𝑎,𝑇𝑎))𝑥,𝑎))≤max𝑥,𝑏)),𝜌𝑏,𝑎))≤𝜌𝑎,𝑇𝑎)) (2.5) So for every 𝐵_𝑎∈ℰ₁, 𝐵_𝑏⊆𝐵_𝑎; that is, 𝐵_𝑏 is an upper bound in ℰ for the family ℰ₁. By Zorn's lemma, there exists a maximal element in ℰ₁, say 𝐵_𝑧. If 𝑏∈𝐵_𝑧, 𝜌(𝑑(𝑏,𝑧))≤𝜌(𝑑(𝑧,𝑇𝑧)), and we get 𝜌(𝑑(𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑(𝜌(𝑑((𝑑(()𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑((𝑑(.𝑏,𝑇𝑏))≤max𝑏,𝑧)),𝜌𝑧,𝑇𝑧)),𝜌𝑇𝑧,𝑇𝑏))<max𝑏,𝑧)),𝜌𝑧,𝑇𝑧)),max𝑧,𝑇𝑧)),𝜌𝑏,𝑇𝑏))by2.1≤max𝑏,𝑧)),𝜌𝑧,𝑇𝑧)),𝜌𝑏,𝑇𝑏))≤max𝑧,𝑇𝑧)),𝜌𝑏,𝑇𝑏))=𝜌𝑧,𝑇𝑧)) (2.6) Then 𝜌(𝑑(𝑏,𝑇𝑏))≤𝜌(𝑑(𝑧,𝑇𝑧)). (2.7) Since 𝑏∈𝐵_𝑏∩𝐵_𝑧, we have 𝐵_𝑏⊆𝐵_𝑧 by Lemma 1.4. But 𝑇𝑏∈𝐵_𝑏, so 𝑇∶𝐵_𝑧→𝐵_𝑧. Now we show that 𝜌(𝑓(𝑏))=𝜌(𝑓(𝑧)) for every 𝑏∈𝐵_𝑧. It is clear that 𝜌(𝑑(𝑏,𝑇𝑏))≤𝜌(𝑑(𝑧,𝑇𝑧)), for all 𝑏∈𝐵_𝑧. Suppose 𝜌(𝑑(𝑏,𝑇𝑏))<𝜌(𝑑(𝑧,𝑇𝑧)) for some 𝑏∈𝐵_𝑧. We have 𝜌(𝑑(𝑏,𝑧))≤𝜌(𝑑(𝑧,𝑇𝑧)), and 𝜌(𝑑(𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑(𝜌(𝑑((𝑑(()𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑((𝑑(.𝑧,𝑇𝑧))≤max𝑧,𝑏)),𝜌𝑏,𝑇𝑏)),𝜌𝑇𝑏,𝑇𝑧))<max𝑏,𝑧)),𝜌𝑏,𝑇𝑏)),max𝑏,𝑇𝑏)),𝜌𝑧,𝑇𝑧))by2.1≤max𝑏,𝑧)),𝜌𝑏,𝑇𝑏)),𝜌𝑧,𝑇𝑧))≤max𝑏,𝑇𝑏)),𝜌𝑧,𝑇𝑧))=𝜌𝑧,𝑇𝑧)) (2.8) which is a contradiction. Thus 𝑓 is modular locally constant on 𝐵_𝑧.
Suppose that (2.2) holds. As above, let 𝑏∈Ω and 𝐵_𝑎∈ℰ₁. Obviously 𝑏∈𝐵_𝑎, so 𝜌(𝑑(𝑎,𝑏))≤𝜌(𝑑(𝑎,𝑇𝑎)). For any 𝑥∈𝐵_𝑏, we have𝜌(𝑑((𝑑(𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑(()(𝑑(.𝑥,𝑏))≤𝜌𝑏,𝑇𝑏))≤max𝑏,𝑎)),𝜌𝑎,𝑇𝑎)),𝜌𝑇𝑎,𝑇𝑏))≤max𝑏,𝑎)),𝜌𝑎,𝑇𝑎))by2.2=𝜌𝑎,𝑇𝑎)) (2.9) Thus 𝜌(𝑑(𝜌(𝑑((𝑑((𝑑(,𝜌(𝑑(𝜌(𝑑((𝑑((𝑑(.𝑥,𝑎))≤max𝑥,𝑏)),𝜌𝑏,𝑎))≤𝜌𝑎,𝑇𝑎))𝑥,𝑎))≤max𝑥,𝑏)),𝜌𝑏,𝑎))≤𝜌𝑎,𝑇𝑎)) (2.10) So, for every 𝐵_𝑎∈ℰ₁, 𝐵_𝑏⊆𝐵_𝑎; that is, 𝐵_𝑏 is an upper bound for the family ℰ₁. Again, by Zorn's lemma there exists a maximal element in ℰ₁, say 𝐵_𝑧. For any 𝑏∈𝐵_𝑧, we have 𝜌(𝑑(𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑((𝑑(()(𝑑(.𝑏,𝑇𝑏))≤max𝑏,𝑧)),𝜌𝑧,𝑇𝑧)),𝜌𝑇𝑧,𝑇𝑏))≤max𝑏,𝑧)),𝜌𝑧,𝑇𝑧)),𝜌𝑧,𝑏))by2.2=𝜌𝑧,𝑇𝑧)) (2.11) This implies that 𝑏∈𝐵_𝑏∩𝐵_𝑧, and Lemma 1.4 gives 𝐵_𝑏⊆𝐵_𝑧. Since 𝑇𝑏∈𝐵_𝑏, so 𝑇∶𝐵_𝑧→𝐵_𝑧.
If 𝑧=𝑇𝑧, then 𝑓(𝑥)=0 on 𝐵_𝑧 and this yields the result. If 𝑧≠𝑇𝑧, we show that 𝜌(𝑓(𝑏))=𝜌(𝑓(𝑧)) for every 𝑏∈𝐵_𝑧. Since 𝜌(𝑑(𝑏,𝑇𝑏))≤𝜌(𝑑(𝑧,𝑇𝑧)) for any 𝑏∈𝐵_𝑧, let us suppose that for some 𝑏∈𝐵_𝑧, 𝜌(𝑑(𝑏,𝑇𝑏))<𝜌(𝑑(𝑧,𝑇𝑧)). So 𝜌(𝑑(𝑏,𝑧))≤𝜌(𝑑(𝑧,𝑇𝑧)) and𝜌(𝑑(𝜌(𝑑((𝑑((𝑑(𝜌(𝑑((𝑑((𝑑(()(𝑑(,𝑧,𝑇𝑧))≤max𝑧,𝑏)),𝜌𝑏,𝑇𝑏)),𝜌𝑇𝑏,𝑇𝑧))≤max𝑏,𝑧)),𝜌𝑏,𝑇𝑏)),𝜌𝑧,𝑏))by2.2=𝜌𝑏,𝑧)) (2.12) thus 𝜌(𝑑(𝑏,𝑧))=𝜌(𝑑(𝑧,𝑇𝑧)). But 𝜌(𝑑(𝑏,𝑧))=𝜌(𝑑(𝑧,𝑇𝑧))>𝜌(𝑑(𝑏,𝑇𝑏)) implies that 𝑧∈𝐵_𝑧, but 𝑧∉𝐵_𝑏 and hence 𝐵_𝑏⊊𝐵_𝑧which contradicts the maximality of 𝐵_𝑧. This completes the proof.

In the following, we assume that (𝒳,𝑑,𝒫) is a spherically complete unital-normal vector ultrametric space.

Corollary 2.2. Let 𝑇∶𝒳→𝒳 be a mapping such that for all 𝑥,𝑦∈𝒳,𝑥≠𝑦, 𝜌𝑑𝑇𝑥,𝑇𝑦<max𝜌𝑑𝑦,𝑇𝑥,𝜌𝑑𝑥,𝑇𝑦. (2.13) Then there exists a subset 𝐵 of 𝒳 such that 𝑇∶𝐵→𝐵 and the mapping 𝑓 defined in (2.3) is modular locally constant.

Proof. Since 𝜌𝑑𝜌𝑑(𝑑(,𝜌𝑑𝜌𝑑𝑑,𝑦,𝑇𝑥≤max𝑦,𝑥,𝜌𝑥,𝑇𝑥))𝑥,𝑇𝑦≤max𝑥,𝑦,𝜌𝑦,𝑇𝑦 (2.14) for all 𝑥,𝑦∈𝒳,𝑥≠𝑦, we get 𝜌𝑑𝑥,𝑦≤max𝜌(𝑑(𝑥,𝑇𝑥)),𝜌𝑑𝑇𝑥,𝑇𝑦,𝜌𝑑𝑇𝑦,𝑦 (2.15) for all 𝑥,𝑦∈𝒳,𝑥≠𝑦. Now, if max𝜌(𝑑(𝑥,𝑇𝑥)),𝜌𝑑𝑦,𝑇𝑦<𝜌𝑑𝑇𝑥,𝑇𝑦, (2.16) then 𝜌𝑑𝜌𝑑𝑑()𝜌𝑑(𝑑(𝑑()𝜌(𝑑(𝑑𝑑𝑑,()𝑇𝑥,𝑇𝑦<max𝑦,𝑇𝑥,𝜌𝑥,𝑇𝑦by2.13≤max𝑥,𝑦,𝜌𝑥,𝑇𝑥)),𝜌𝑦,𝑇𝑦by2.14≤max𝑥,𝑇𝑥)),𝜌𝑇𝑥,𝑇𝑦,𝜌𝑇𝑦,𝑦=𝜌𝑇𝑥,𝑇𝑦by2.16 (2.17) which is a contradiction. Thus 𝜌(𝑑(𝑇𝑥,𝑇𝑦))≤max{𝜌(𝑑(𝑥,𝑇𝑥)),𝜌(𝑑(𝑦,𝑇𝑦))}, and so 𝜌𝑑𝑥,𝑦≤max𝜌(𝑑(𝑥,𝑇𝑥)),𝜌𝑑𝑦,𝑇𝑦. (2.18) Therefore 𝜌𝑑𝜌𝑑𝑑,()𝜌𝑑(𝑑(𝑑()𝜌(𝑑(𝑑.()𝑇𝑥,𝑇𝑦<max𝑦,𝑇𝑥,𝜌𝑥,𝑇𝑦by2.13≤max𝑥,𝑦,𝜌𝑥,𝑇𝑥)),𝜌𝑦,𝑇𝑦by2.14≤max𝑥,𝑇𝑥)),𝜌𝑦,𝑇𝑦by2.18 (2.19) Now, Theorem 2.1 completes the proof.

Corollary 2.3. Let 𝑇∶𝒳→𝒳 be a mapping such that for all 𝑥,𝑦∈𝒳,𝑥≠𝑦, 𝜌𝑑𝑇𝑥,𝑇𝑦<𝜌𝑑𝑥,𝑦. (2.20) Then there exists a subset 𝐵 of 𝒳 such that 𝑇∶𝐵→𝐵 and the mapping 𝑓 defined in (2.3) is modular locally constant.

Proof. We have 𝜌𝑑𝜌(𝑑(𝑑𝑑𝜌𝑑(𝑑(𝑑()𝜌(𝑑(𝑑,𝑥,𝑦≤max𝑥,𝑇𝑥)),𝜌𝑇𝑥,𝑇𝑦,𝜌𝑇𝑦,𝑦<max𝑥,𝑦,𝜌𝑥,𝑇𝑥)),𝜌𝑇𝑦,𝑦by2.20≤max𝑥,𝑇𝑥)),𝜌𝑦,𝑇𝑦 (2.21) for all 𝑥,𝑦∈𝒳,𝑥≠𝑦. Again, Theorem 2.1, completes the proof.

Acknowledgments

The authors would like to thank the referee for his/her valuable comments on this paper. The second author’s research was in part supported by a grant from IPM (No. 89470128).