A new version of the results of U n - hypermetric spaces

: Introduction/purpose: e aim of this paper is to present the concept of a universal hypermetric space. An n-dimensional (n ≥ 2) hypermetric distance over an arbitrary non-empty set X is generalized. is hypermetric distance measures how separated all n points of the space are. e paper discusses the concept of completeness, with respect to this hypermetric as well as the ﬁxed point theorem which play an important role in applied mathematics in a variety of ﬁelds. Methods: Standard proof based theoretical methods of the functional analysis are employed. Results: e concept of a universal hypermetric space is presented. e universal properties of hypermetric spaces are described. Conclusion: is new version of the results for U n -hypermetric spaces may have applications in various disciplines where the degree of clustering is sought for


Introduction
e role of distance in understanding the world is undeniable.Our intuitive understanding of the concept of distance in the real world, however, is different from the one proposed in mathematics.Some of the properties that belong to our understanding of distance from the real world, such as symmetry and singlevaluedness, are not necessarily established within certain abstract distances.
is will, in fact, be our main motivation for presenting a generalized concept of distance as a set-valued function in this paper.e notion of 2-metric spaces, as a possible generalization of metric spaces, was introduced by Gähler (Gähler, 1963), (Gähler, 1964), (Gähler, 1966).See also (Diminnie et al, 2017), (Ha et al, 1990) for further developments.e 2-metric d(x, y, z) is a function of 3 variables, and Gähler geometrically interpreted it as an area of a triangle with the vertices at x , y and z, respectively.is led B. C. Dhage, in his PhD thesis in 1992, to introduce the notion of D-metric (Dhage et al, 2000) that does, in fact, generalize metric spaces.Subsequently, Dhage published a series of papers attempting to develop topological structures in such spaces and prove several fix point results.
Most of the claims, however, concerning the fundamental topological properties of D-metric spaces, are incorrect.In 2003, Mustafa and Sims demonstrated that in a strong remark (Mustafa & Sims, 2003).is led them to introducing the notion of a G-metric space (Mustafa & Sims, 2006), as a generalization of metric spaces.In this type of spaces, a non-negative real number is assigned to every triplet of elements.
In an attempt to generalize the notion of a G-metric space to more than three variables, Khan first introduced the notion of a K-metric, and later the notion of a generalized n-metric space (for any n ≥ 2) (Khan, 2012), (Khan, 2014).He also proved a common fixed point theorem for such spaces.
e main purpose of this paper is a generalization of universal metric spaces into universal hypermetric spaces of the n-dimension (see (Kelly, 1975) for a discussion on hypermetric spaces).In the first part, we generalize an n-dimensional (n ≥ 2) hypermetric distance over an arbitrary non-empty set X. is hypermetric measures how separated all n points of the space are.e hyperdistance function is defined in any way we like, the only constraint being the simultaneous satisfaction of the three properties, viz non-negativity and positive-definiteness, symmetry and triangle inequality.In the second part, we discuss the concept of completness, with respect to this hypermetric, and the fixed point theorem, which play an important role in applied mathematics in a variety of fields.Examples show a fundamental difference between our results and the well-known ones.is concept is the first view of novel methods for selecting the clusters by hypermetric.e purpose definition is applicable for engineering science (for example, the theory of clustering).
By a strict order relation of a set X, we mean a binary relation " < ", which is transitive (α < β and β < γ implies α < γ), such that α < β and β < α cannot both hold.It is a strict total order relation, if for every α,β belonging to X, exactly one and only one of α < β, β < α or α = β holds.A group G is called le-ordered, if endowed with a strict total relation " < " which is le invariant, meaning that α < β implies γ + α < γ + β, for all α,β,γ ∈ G.We will say that G is bi-ordered, if it admits the le and right invariant properties simultaneously (historically, this has been called simple-ordered).We refer to the ordered pair (G, <) as an ordered group (Cohen & Goffman, 1949).From now on, we assume that 1 denotes the identity element of G.It should be noted that, for abelian additive groups, the identity element may be denoted by 0. is is common to an ordered group with the symbol " ≤ " that has the obvious meaning : α ≤ β means α < β or α = β.We denote G + a set of non-negative elements of G, namely .Two positive elements, x, y, of an ordered group are relatively Archimedean if there are positive integers m, n such that mx ≥ y and ny ≥ x.If every two positive elements of an ordered group are relatively Archimedean, then the ordered group is Archimedean.
Every Archimedean ordered group is isomorphic to an ordered subgroup of the additive group of the real numbers.An ordered group G is order complete if every non-empty subset of G that has an upper bound has a least upper bound.
Universal hypermetric spaces of the dimension n e goal of this section is to describe a few properties of the universal hypermetric spaces.
Definition 1.Let G be an ordered group.An ordered group metric (or OG-metric ) on a non-empty set X is a symmetric non-negative function d G om X × X into G such that d G (x, y) = 0 if and only if x = y and such that the triangle inequality is satisfied; the pair (X, d G ) is an ordered group metric space (or OG-metric space).
Now we first recall and introduce some notation.For n ≥ 2, let Xn denote the n-times Cartesian product and G be an ordered group.Let P*(G) denote the family of all non-empty subsets of G.We begin with the following definition.
Definition 2. Let X be a non-empty set.Let : X n → P*(G + ) be a function that satisfies the following conditions: Let A i be subsets of X, i = 1, . . ., n.We define We will use the following abbreviated notation: e function is called a universal ordered hypermetric group of the dimension n, or more specifically an UO n -hypermetric (or U n -hypermetric) on X, and the pair (X, ) is called an U n -hypermetric space.For example, we can set G + = or , where := ∪ {0} = {0, 1, 2, . . .} and := [0, +∞).
In the sequel, for simplicity we assume that G+ = .e following useful properties of a Un-hypermetric are easily derived from the axioms.Proposition ) be a U n -hypermetric space, then for any x 1 , ..., x n , a # X it follows that: Proposition 3. Let (X, ) be a Un-hypermetric space, then {0} # (x 1 , ..., x n ) for all x 1 , ..., x n # X. Proof.By the condition (U4) of the definition of a U n -hypermetric space, we have {0} = (x 1 , ..., x n ) # (x 1 , ..., x n ).Proposition 4. Every Un-hypermetric space (X, ) defines a U 2 -hypermetric space (X, ) as follows: Proposition 5. Let e be an arbitrary positive real value number, and (X, d) be a metric space.We define an induced hypermetric, en (X, ) is a U 2 -hypermetric space

Main results
Let (X, ) be a U n -hypermetric space and be a partition of X.For each point p ∈ X, we denote a point in containing p, and we denote the equivalent relation induced by the relation by # Definition 3. Let (X, ) be a U n -hypermetric space.Let p 1 , . . ., p n # X, and consider .A quotient U n -hypermetric of the points of induced by is the function given by .Proposition 6. e quotient U n -hypermetric induced by is well-defined and is a U n -hypermetric on .
Let (X, ) be a U n -hypermetric space of a dimension n > 2. For any arbitrary a in X, define the function on X n−1 by en we have the following result.Proposition 7. e function defines a U n−1 -hypermetric on X. Proof.We will verify that satisfies the five properties of a U n−1 -hypermetric.Proposition 8. Let f : X → Y be an injection om a set X to a set Y .If : X n → P* is a U n -hypermetric on the set Y , then : X n → P* , given by the formula (x 1 , . . ., x n ) = (f(x 1 ), . . ., f(x n )) for all x 1 , . . ., x n # X, is a U n -hypermetric on the set X.
Proposition 9. Let (X, ) be any U n -hypermetric space.Let λ be any positive real number.en (X, ) is also a U n -hypermetric space where So, on the same X many U n -hypermetrics can be defined, as a result of the procedure in which the same set X is endowed with different metric structures.Another structure in the next proposition is useful for scaling the U n -hypermetric, so we need the following explanation.
For any non-empty subset A of , and λ ∈ R + we define a set λ.A to be .Proposition 10.Let (X, ) be any U n -hypermetric space.Let Λ be any positive real number.We define (x 1 , . . ., x n ) = λ.(x 1 , . . ., x n ).en (X, ) is also a U n -hypermetric space.
A sequence {x m } in a U n -hypermetric space (X, ) is said to converge to a point s in X, if for any > 0 there exists a natural number N such that for every m 1 , . . ., m n−1 ≥ N, then we write, Definition 12. Let (X, ) and (Y, ) be two U n -hypermetric spaces.A function f : X −→ Y is called a U ncontraction if there exists a constant k # [0, 1) such that (f(x 1 ), ..., f(x n )) = k (x 1 , ..., x n ) for all x 1 , . . ., x n # X.
Proof.We consider x m+1 = T(x m ), with x 0 being any point in X.We have, by repeated use of the rectangle inequality and the application of contraction property, the following: for all m, s 1 ∈ which m < s 1 and k ∈ [0, 1).en we have since For m ≤ s 1 ≤ s 2 ∈ and (U5) implies that now taking the limit as m, s 1 , s 2 → +∞, we get Now for m ≤ s 1 ≤ s 2 ≤ . . .≤ s n−1 ∈ , we will have then {x m } is a Cauchy sequence.By completeness of (X, ), there exists a ∈ X such that {x n } isconvergent to a. e fact that the limit x m is a fixed point of T follows the -continuity of T, and

Conclusions
In this article, we have put forward a development of the results of U n -hypermetric spaces, covering a variety of topics relevant for understanding their properties including completeness and the fixed-point theorem.We believe this work may be relevant from both the theoretical standpoint and the point of view of applications in contemporary problems such as those of clusterings which oen appear in practice.

Finally
, if a and b are two fixed points, then Since k < 1, we have (a, b, . . ., b) = {0}, so a = b and the fixed point is unique.