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Research article Special Issues

q-rung logarithmic Pythagorean neutrosophic vague normal aggregating operators and their applications in agricultural robotics

  • Received: 18 April 2023 Revised: 18 August 2023 Accepted: 11 September 2023 Published: 07 November 2023
  • MSC : 06D72, 90B50

  • The article explores multiple attribute decision making problems through the use of the Pythagorean neutrosophic vague normal set (PyNVNS). The PyNVNS can be generalized to the Pythagorean neutrosophic interval valued normal set (PyNIVNS) and vague set. This study discusses q-rung log Pythagorean neutrosophic vague normal weighted averaging (q-rung log PyNVNWA), q-rung logarithmic Pythagorean neutrosophic vague normal weighted geometric (q-rung log PyNVNWG), q-rung log generalized Pythagorean neutrosophic vague normal weighted averaging (q-rung log GPyNVNWA), and q-rung log generalized Pythagorean neutrosophic vague normal weighted geometric (q-rung log GPyNVNWG) sets. The properties of q-rung log PyNVNSs are discussed based on algebraic operations. The field of agricultural robotics can be described as a fusion of computer science and machine tool technology. In addition to crop harvesting, other agricultural uses are weeding, aerial photography with seed planting, autonomous robot tractors and soil sterilization robots. This study entailed selecting five types of agricultural robotics at random. There are four types of criteria to consider when choosing a robotics system: robot controller features, cheap off-line programming software, safety codes and manufacturer experience and reputation. By comparing expert judgments with the criteria, this study narrows the options down to the most suitable one. Consequently, q has a significant effect on the results of the models.

    Citation: Murugan Palanikumar, Chiranjibe Jana, Biswajit Sarkar, Madhumangal Pal. q-rung logarithmic Pythagorean neutrosophic vague normal aggregating operators and their applications in agricultural robotics[J]. AIMS Mathematics, 2023, 8(12): 30209-30243. doi: 10.3934/math.20231544

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  • The article explores multiple attribute decision making problems through the use of the Pythagorean neutrosophic vague normal set (PyNVNS). The PyNVNS can be generalized to the Pythagorean neutrosophic interval valued normal set (PyNIVNS) and vague set. This study discusses q-rung log Pythagorean neutrosophic vague normal weighted averaging (q-rung log PyNVNWA), q-rung logarithmic Pythagorean neutrosophic vague normal weighted geometric (q-rung log PyNVNWG), q-rung log generalized Pythagorean neutrosophic vague normal weighted averaging (q-rung log GPyNVNWA), and q-rung log generalized Pythagorean neutrosophic vague normal weighted geometric (q-rung log GPyNVNWG) sets. The properties of q-rung log PyNVNSs are discussed based on algebraic operations. The field of agricultural robotics can be described as a fusion of computer science and machine tool technology. In addition to crop harvesting, other agricultural uses are weeding, aerial photography with seed planting, autonomous robot tractors and soil sterilization robots. This study entailed selecting five types of agricultural robotics at random. There are four types of criteria to consider when choosing a robotics system: robot controller features, cheap off-line programming software, safety codes and manufacturer experience and reputation. By comparing expert judgments with the criteria, this study narrows the options down to the most suitable one. Consequently, q has a significant effect on the results of the models.



    A metric space M=(X,d) is formed by a set X of points and a distance d defined in X. In particular, the vertex set of any connected graph, equipped with the shortest path distance, is a metric space. This suggests approaching problems related to metric parameters in graphs as more general problems in the context of metric spaces. In some cases, a problem has been formulated within the theory of metric spaces and has become popular and extensively studied in graph theory. This is the case of the metric dimension: a theory suggested by Blumenthal [3] in 1953 in the general context of metric spaces that became popular more than twenty years later, after being introduced by Slater [15,16] in 1975 and by Harary and Melter [8] in 1976, in the particular context of graph theory. Some graph operations have been extensively studied and then generalized to the case of metric spaces. For instance, the notion of the join of two graphs was extended to metric spaces by Beardon and Rodríguez-Velázquez [2], while the notion of the lexicographic product of two graphs was extended to metric spaces by Rodríguez-Velázquez [12]. In other cases, like the Cartesian product of graphs, and the Cartesian product of metric spaces, the theory has been widely studied in both contexts.

    In this paper, we consider the study of corona metric spaces as a natural generalization of the theory of corona product graphs. In particular, we define the concept of a corona metric space in such a way that the set of vertices of any corona product graph, equipped with the shortest path distance, is a corona metric space. We study some basic properties of corona metric spaces, such as completeness, boundedness, compactness and separability. Furthermore, we provide necessary and sufficient conditions for the existence of universal lines in these metric spaces, and we obtain a formula for the metric dimension.

    The remainder of the article is structured as follows. In Section 2 we define the concept of a corona metric space, and we also justify the terminology used. In Section 3 we briefly describe the notion of a gravitational metric space, since the properties of these spaces are closely linked to the properties of corona metric spaces. Section 4 is devoted to the study of basic properties of these metric spaces. In Section 5 we study the metric dimension, while in Section 6 we discuss the existence of universal lines.

    We assume that the reader is familiar with the basic concepts and terminology of metric spaces. If this is not the case, we suggest the textbooks [10,11].

    In this section we will establish the notation and terminology related to corona metric spaces. Given a metric space M=(X,d) and a point xX, we define the nearness of x to be

    η(x)=inf{d(x,y):yX{x}}.

    We define the nearness of the metric space to be

    η(M)=inf{η(x):xX}.

    Notice that the vertex set V(G) of any connected graph G, equipped with the shortest path distance, is a metric space with nearness equal to one.

    Next, we introduce the concept of a corona metric space.

    Definition 2.1. Let M=(X,dX) be a metric space with 0<η(M)<+ and M be a family of pairwise disjoint metric spaces such that there exists a bijection f:XM. Let f(x)=Mx=(Xx,dXx) for every xX, and

    U=X(xXXx).

    Let dU:U×UR be the map defined by

    dU(a,b)={dX(a,b),ifa,bX,η(x)+dX(x,x),ifaXxandb=xX,η(x)+dX(x,x)+η(x),ifaXxandbXx,min{2η(x),dXx(a,b)},ifa,bXx.

    The map dU is called a corona distance, and the metric space MfM=(U,dU) is called a corona metric space.

    One can easily check that dU is a distance function. For this, by the definition of dU, it is only necessary to check the triangular inequality dU(x,z)dU(x,y)+dU(y,z) by differentiating cases according to the positions of x,y and z in the set U. We omit the details.

    Notice that the requirement η(M)<+ is needed, as inf=+. Hence, if |X|=1, then η(M)=+. Therefore, the requirement η(M)<+ is equivalent to |X|2. Moreover, when defining dU, we really could have used another positive constant instead of η(x). The convenience of using η(x) lies in keeping the points of Mx at the same distance from x as the closest point to x in X, by analogy with what happens in graph theory for the case of corona graphs.

    The name corona metric space is inherited from graph theory. Let us briefly describe this idea. Let G be a connected graph, H be a family of pairwise disjoint graphs with |H|=|V(G)| and g:V(G)H be a bijection. Let g(x)=Hx for every xV(G). We define the generalized corona graph GgH as the graph obtained from G, H and g, by taking one copy of G and one copy of each graph in H, and making x adjacent to every vertex of g(x)=Hx for every xV(G). Notice that GgH is connected if and only if G is connected. Now, if there exists a graph H such that g(x)=Hx is isomorphic to H for every xV(G), then GgHGH is a standard corona graph, as defined in 1970 by Frucht and Harary [7].

    It is not difficult to check that if G is a connected graph, and |V(G)|2, then the metric space induced by any generalized corona graph GgH is a corona metric space MfM, thus justifying the terminology used. To see this, we only need to take M=(X,dX)=(V(G),dG) and f(x)=Mx=(Xx,dXx)M for every xV(G), where Xx=V(Hx) and dXx(y,y)=min{2,dHx(y,y)} for every y,yV(Hx). As we can expect, if there exists xV(G) such that the graph g(x)=Hx is not connected, then we assume that dHx(y,y)=+ for every pair of vertices y and y belonging to different components of Hx, and so dXx(y,y)=min{2,dHx(y,y)}=min{2,+}=2.

    In this section we will establish the notation and terminology related to gravitational metric spaces. By the definition of a corona metric space, for any xX, the metric subspace Mx=(Xx,dx) of MfM, induced by the set Xx, is equipped with the distance function

    dx(a,b)=min{2η(x),dXx(a,b)}foreverya,bXx.

    These metric spaces play an important role in the study of join metric spaces [2] and in the study of lexicographic metric spaces [12]. In fact, as described in [12], the metric space Mx is isometric to a metric space obtained from Mx, which can be seen as a deformation of Mx where every point aXx attracts (and is attracted by) every point that is outside the closed ball of center a and radius 2η(x). All these points remain at distance 2η(x) from a in the deformed space. In particular, if the diameter of Mx is at most 2η(x), then Mx is isometric to Mx. The metric space Mx locally, near every point, looks like patches of the space Mx, but the global topology can be quite different.

    In order to preserve the terminology used in [12], we will refer to Mx=(Xx,dx) as the gravitational metric space associated to M=(Xx,dXx) with gravitation constant 2η(x). In fact, it is possible to develop the theory of corona metric spaces with an arbitrary gravitational constant. Obviously, the gravitational constant is always a positive number.

    Furthermore, apart from the theory of corona metric spaces and apart from theory of lexicographic metric spaces developed in [12], the theory of gravitational metric spaces has its own identity, as for any real number t>0 and any metric space M=(X,dX), we can define the gravitational metric space M=(X,d) associated to it, where the distance function d is defined using both, the distance function dX and the gravitational constant t:

    d(a,b)=min{2t,dX(a,b)}foreverya,bX.

    The discrete metric d0 on a non-empty set X is given by d0(x,y)=0 if x=y, and d(x,y)=1 otherwise. In fact, M=(X,d0) is isometric to the metric space induced by a complete graph with vertex set X.

    There are metric spaces with metrics that may differ from the discrete metric but generate the same topology. These metric spaces are known as discrete metric spaces [11]. In order to avoid misunderstandings regarding this concept, we will highlight this fact in the following definition.

    Definition 4.1. [11] A metric space M is called a discrete metric space if and only if all its subsets are open (and therefore closed) in M.

    For instance, the set N of positive integers with its usual metric inherited from the one-dimensional Euclidean space R is a discrete metric space, the vertex set of any graph equipped with the shortest path distance is a discrete metric space, and, in particular, every finite metric space is a discrete metric space.

    Theorem 4.1. [12] Let M=(X,d) be a metric space. If η(M)>0, then M is a complete discrete metric space.

    The next result is a criterion for completeness of any gravitational metric space.

    Theorem 4.2. [12] A metric space is complete if and only if its gravitational metric space, with any gravitational constant, is complete.

    We proceed to show that the completeness of MfM is equivalent to the completeness of all the metric spaces of the family M{M}.

    Theorem 4.3. A corona metric space MfM is complete if and only if M is complete and every MxM is complete.

    Proof. By definition of a corona metric space, η(M)>0, and by Theorem 4.1, M is a complete metric space. Hence, we proceed to show that the corona metric space MfM is complete if and only if every metric space MxM is complete.

    We first assume that MxM is complete for every xX. Let (zn)nN be a Cauchy sequence in MfM. Obviously, we only need to discuss the case in which this sequence is not eventually constant. By the definition of a Cauchy sequence, for every ϵ(0,η(M)) there exists nN such that dU(zn,zm)<ϵ for every n,m>n. Now, by the definition of nearness, we conclude that there exists xX such that znXx for every n>n. Thus, ϵ>dU(zn,zm)=min{2η(x),dXx(zn,zm)}, for every n,m>n. Let (yn)nN be the Cauchy sequence defined by ynn=zn, for every n>n. By Theorem 4.2, the gravitational metric space Mx is complete, and so (yn)nN converges in Mx. Now, since Mx is the subspace of MfM induced by Xx, we conclude that (zn)nN converges in MfM. Therefore, MfM is a complete metric space.

    Conversely, assume that MfM is a complete metric space. Let (yn)nN be a Cauchy sequence in Mx, for MxM. Hence, for every ϵ(0,η(x)) there exists nN such that dU(yn,ym)=dXx(yn,ym)<ϵ for every n,m>n. This implies that (yn)nN is a Cauchy sequence in MfM, which is convergent, by the completeness of MfM. Now, since dU(a,b)η(x)>ϵ for every aXx and bUXx, we can conclude that (yn)nN converges in Mx. Therefore, Mx is a complete metric space, for every MxM.

    The concept of boundedness plays an important role in the theory of metric spaces. Whereas a metric space is bounded if it is included in a single ball, a metric space M=(X,d) is totally bounded if for every ϵ>0 there exists a finite set SX such that X=sSBϵ(s), where

    Bϵ(s)={xX:d(x,s)<ϵ}

    is the ball of center s and radius ϵ. Since the finite union of bounded sets is bounded, every totally bounded space is bounded. In general, the converse is not true.

    Theorem 4.4. [12] A metric space is totally bounded if and only if its gravitational metric space, with any gravitational constant, is totally bounded.

    We proceed to show that the total boundedness of MfM is equivalent to the total boundedness of all the metric spaces of the family M whenever M is a finite metric space.

    Theorem 4.5. A corona metric space MfM, with M=(X,d), is totally bounded if and only if |X|<+ and Mx is totally bounded for every xX.

    Proof. Recall that U=X(xXXx) is the set of points of MfM. Let 0<ϵ<η(M)/2 and SU such that U=sSBϵ(s). Since Bϵ(x)={x} for every xX, Bϵ(y)Bϵ(y)= for every yXx, and yUXx, we can conclude that |S|<+ if and only if (a) |X|<+ and (b) for each xX there exists SxSXx such that |Sx|<+ and Xx=sSxBϵ(s). Notice that (b) is equivalent to saying that the gravitational metric space Mx is totally bounded for every xX. Therefore, by Theorem 4.4 we conclude the proof.

    In order to study the compactness of corona metric spaces, we need to recall the following known theorem, which is a characterization of compact metric spaces, and is a generalization of the Heine-Borel theorem.

    Theorem 4.6. [10] A metric space is compact if and only if it is complete and totally bounded.

    Our next result is a direct consequence of Theorems 4.3, 4.5 and 4.6.

    Theorem 4.7. A corona metric space MfM, with M=(X,d), is compact if and only if |X|<+ and Mx is compact for every xX.

    A set S of points of a metric space M is dense in M if every open ball of M contains at least a point belonging to S. A metric space M is separable if there exists a countable set of points which is dense in M.

    Theorem 4.8. [12] A metric space is separable if and only if its gravitational metric space, with any gravitational constant, is separable.

    Next, we proceed to study the separability of MfM.

    Theorem 4.9. A corona metric space MfM, with M=(X,d), is separable if and only if X is a countable set and Mx is separable for every xX.

    Proof. We proceed to show that MfM is separable if and only if X is a countable set and for any xX the gravitational metric space Mx is separable. After that, the proof is concluded by Theorem 4.8.

    We first assume that X is a countable set, and the gravitational metric space Mx is separable for every xX. Let SxXx be a countable set which is dense in Mx. Notice that S=X(xXSx) is a countable set, as the countable union of countable sets is countable. Now, a ball Bϵ(y) in Mx will be distinguished from a ball Bϵ(y) in MfM by the bold type used. Since Sx is dense in Mx, we can conclude that for any ϵ>0 and yXx,

    Bϵ(y)SBϵ(y)Sx.

    Hence, S=X(xXSx) is dense in MfM, and so MfM is separable.

    Conversely, assume that MfM is separable. Since every subspace of a separable metric space is separable, we can conclude that Mx is separable for every xX. Finally, if W is a countable set which is dense in MfM, then for any xX and yXx, we have XxBη(M)/2(y)W, which implies that |X||W|. Therefore, X is a countable set.

    In the next section we will see that the above results have interesting applications.

    The notion of the metric dimension of a general metric space was introduced for the first time in 1953 by Blumenthal [3] in his book Theory and Applications of Distance Geometry. This theory attracted little attention until, more than twenty years later, it was applied to the particular case of graphs [8,15,16]. Since then, it has been frequently applied and investigated in graph theory and many other disciplines. More recently, in [1,2,9,12], the theory of metric dimension was developed further for general metric spaces. In this section, we study the metric dimension of corona metric spaces.

    Let M=(X,d) be a metric space. If X is an infinite set, we put |X|=+. In fact, it is possible to develop the theory with |X| any cardinal number, but we shall not do this. A subset AX is said to resolve M if for any pair of different points x,yX, there exists a point aA such that d(x,a)d(y,a). Informally, if an object in M knows its distance from each point of A, then it knows exactly where it is located in M. The class R(M) of subsets of X that resolve M is non-empty, since X resolves M. The metric dimension dim(M) of M=(X,d) is defined as

    dim(M)=inf{|A|:AR(M)}.

    The sets in R(M) are called the metric generators of M, and A is a metric basis of M if AR(M) and |A|=dim(M).

    This terminology comes from the fact that a metric generator of a metric space M=(X,d) induces a global co-ordinate system on M. For instance, if (x1,,xr) is an ordered metric generator of M, then the map ψ:XRr given by

    ψ(x)=(d(x,x1),,d(x,xr)) (5.1)

    is injective, which implies that the map ψ is a bijection from X to a subset of Rr, and the metric space inherits its co-ordinates from this subset. As the following result shows, a stronger conclusion arises when M is a compact metric space.

    Theorem 5.1. [2] If M is a compact metric space with dim(M)=r<+, then M is homeomorphic to a compact subspace of the Euclidean spaceRr.

    According to the following result, the boundedness of M affects directly the metric dimension of its gravitational metric spaces.

    Remark 5.2. [6] Let M be a metric space and M be its gravitational metric space for an arbitrary gravitational constant. If M is unbounded, then dim(M)=+.

    In general, the converse of Remark 5.2 does not hold. To see this, it is enough to consider M=(R,d0), where d0 is the discrete metric. In such a case, dim(M)=+ and M is bounded.

    Theorem 5.3. Let MfM be a corona metric space. If X is the set of points of M, and every metric space in M has at least two points, then

    dim(MfM)=xXdim(Mx).

    Proof. Notice that, by the definition of a corona metric space, 0<η(M)<+, and so |X|2. Let W be a metric basis of MfM and let Wx=WXx for every xX. Since for every u,vXx and every aUXx

    dU(a,u)=dX(a,x)+η(x)=dU(a,v),

    we deduce that Wx is a metric generator of the subspace of MfM induced by Xx, which coincides with the gravitational metric space Mx, and so |Wx|dim(Mx). Hence, if dim(Mx)=+ for some xX, or |X|=+, then dim(MfM)=+, and the result follows. From now on, assume that |X|<+ and dim(Mx)<+ for every xX. Thus,

    dim(MfM)=|W|xX|Wx|xXdim(Mx).

    It remains to show that

    dim(MfM)xXdim(Mx).

    To this end, we proceed to show that if Ax is a metric basis of Mx for every xX, then A=xXAx is a metric generator for MfM. We differentiate the following cases for u,vUA, where uv.

    Case 1. u,vXx, for some xX. Since Ax is a metric basis of Mx=(Xx,dx), there exists aAx such that

    dU(a,u)=min{2η(x),dXx(a,u)}=dx(a,u)dx(a,v)=min{2η(x),dXx(a,v)}=dU(a,v).

    Case 2. uX and vXu. In this case, for every xX{u} and aAx,

    dU(a,v)=η(x)+dX(x,u)+η(u)>dX(x,u)+η(x)=dU(a,u).

    Case 3. uX and vXx, where xX{u}. For any aAu,

    dU(a,u)=η(u)<η(u)+dX(u,x)+η(x)=dU(a,v).

    Case 4. uXx and vXx, for some pair of different points x,xX. Since η(x)dX(x,x), for any aAx we deduce that

    dU(a,u)=min{2η(x),dXx(a,u)}<η(x)+dX(x,x)+η(x)=dU(a,v).

    Case 5. u,vX. In this case, for every aAu we have

    dU(a,u)=η(u)<η(u)+dX(u,v)=dU(a,v).

    According to the five cases above, A is a metric generator of MfM, and so

    dim(MfM)|A|=xX|Ax|=xXdim(Mx).

    Therefore, the result follows.

    The following result is a direct consequence of Theorem 5.3.

    Corollary 5.4. Let MfM be a corona metric space such that every metric space in M has at least two points. If |M|=+, or there exists MxM such that dim(Mx)=+, thendim(MfM)=+.

    By combining Theorems 4.7, 5.1 and 5.3, we deduce the following result.

    Theorem 5.5. Let MfM be a corona metric space such that |M|<+, every metric space MxM has at least two points, and dim(Mx)<+. Let m=MxMdim(Mx). If every MxM is a compact metric space, then MfM is homeomorphic to a compact subspace of the Euclidean space Rm.

    We conclude this section with the following theorem, which is another interesting result on compact metric spaces.

    Theorem 5.6. Let MfM be a corona metric space such that every metric space MxM has at least two points. If MfM is a compact metric space, and dim(MfM)<+, then the following statements hold.

    (ⅰ) For any MxM, the gravitational metric space Mx is homeomorphic to a compact subspace of the Euclidean space Rmx, where mx=dim(Mx).

    (ⅱ) Every metric space MxM is homeomorphic to a compact subspace of the Euclidean space Rnx, where nx=dim(Mx)dim(Mx).

    Proof. Let MxM and dim(Mx)=mx. By Theorems 4.2, 4.4 and 4.6, we have that Mx is compact if and only if Mx is compact. On the other hand, if MfM is compact, then Mx is compact, by Theorem 4.7. In summary, from the compactness of MfM, we infer the compactness of Mx. Now, if dim(MfM)<+, then Theorem 5.3 leads to dim(Mx)<+. Therefore, by Theorem 5.1 we complete the proof of (ⅰ).

    In order to prove (ⅱ), notice that if a point distinguishes two elements in Mx, it also distinguishes them in Mx, which implies that dim(Mx)dim(Mx)<+. Finally, since Mx is compact, by Theorem 5.1 we complete the proof.

    Let M=(X,dX) be a metric space. Given two different points x,xX, we define

    [x,x]={yX:dX(x,x)=dX(x,y)+dX(y,x)}.

    A line LM{x,x} induced by two distinct points x,xX is defined by

    LM{x,x}={yX:y[x,x]orx[y,x]orx[x,y]}.

    A line LM{x,x} is universal whenever LM{x,x}=X.

    The following conjecture was stated by Chen and Chvátal [4] for the case of finite metric spaces, where (M) denotes the number of distinct lines in M.

    Conjecture 6.1. (The Chen-Chvátal conjecture, [4]) Every finite metric space M=(X,dX) with at least two points either has a universal line or (M)|X|.

    The Chen-Chvátal conjecture is an attempt to generalize the de Bruijn-Erdős theorem of Euclidean geometry, which asserts that every noncollinear set of n points in the plane determines at least n distinct lines.

    The Chen-Chvátal conjecture remains open, and, as time goes by, the interest of mathematicians in solving it is increasing. Recently, Chvátal [5] described the main progress related to the conjecture and pointed out twenty-nine related open problems plus three additional conjectures. As one might expect, the paper highlights the existence of some families of metric spaces for which the conjecture has been solved.

    Recently, Rodríguez-Velázquez [13] has solved the conjecture for some particular classes of metric spaces. Among other results, he has shown that if MfM=(U,dU) is a finite corona metric space, where η(x)=1 for every point x of M, then either MfM has a universal line or (MfM)|U|.

    The problem of investigating the properties of (finite or infinite) metric spaces having a universal line is interesting. In the case of a finite metric space M=(X,dX), the problem remains interesting even if we can check that (M)|X|. Recently, Rodríguez-Velázquez [14] discussed the problem for the particular case of metric spaces induced by graphs, including the case of metric spaces induced by corona graphs. In this section we analyse the problem for the general class of not necessarily finite corona metric spaces.

    To begin with, we need to introduce the following notation. The diameter of a metric space M=(X,dX) is defined to be

    D(M)=sup{dX(x,y):x,yX}.

    Theorem 6.2. Let MfM be a corona metric space and X the set of points of M. Then MfM has a universal line if and only if at least one of the following conditions holds.

    (ⅰ) There exist xX and yXx such that either Xx={y} or dXx(y,z)2η(x) for every zXx{y}.

    (ⅱ) There exist two different points x,xX such that [x,x]={x,x}, and the line LM{x,x} is universal.

    Proof. Observe that, by Definition 2.1, |X|2, as 0<η(M)<+. First, assume that (ⅰ) holds.

    If there exists xX such that |Xx|=1, say Xx={y}, then for any zU{x,y}, we have that x[y,z], as

    dU(y,z)=η(x)+dU(x,z)=dU(y,x)+dU(x,z).

    Therefore, the line LMfM{x,y} is universal.

    On the other hand, if there exists yXx such that dXx(y,z)2η(x) for every zXx{y}, then

    dU(z,y)=min{2η(x),dXx(y,z)}=2η(x)=dU(z,x)+dU(x,y),

    while for every zU(XX{x}) we have that x[y,z], as

    dU(y,z)=η(x)+dU(x,z)=dU(y,x)+dU(x,z).

    Therefore, the line LMfM{x,y} is universal.

    Now, assume that (ⅱ) holds. Let x,xX be two different points such that [x,x]={x,x} and the line LM{x,x} is universal. In this case, for any xX, either x[x,x] or x[x,x], which implies that for any z{x}Xx, either x[x,z] or x[x,z]. Therefore, the line LMfM{x,x} is universal.

    Conversely, assume that there exist two points u,vU such that the line LMfM{u,v} is universal. We differentiate the following cases.

    Case 1. u,vX. Since XLMfM{u,v}, we conclude that the line LM{u,v} is universal. Now, suppose that there exists z[u,v]{u,v}. Since for any wXz,

    dU(u,w)+dU(w,v)=dX(u,z)+dX(z,v)+2η(z)>dX(u,z)+dX(z,v)=dU(u,v),

    and also

    |dU(u,w)dU(w,v)|=|dX(u,z)dX(z,v)|<dX(u,z)+dX(z,v)=dU(u,v),

    we obtain that wLMfM{u,v}, which is a contradiction. Therefore, (ⅱ) follows.

    Case 2. u,vXx for some point xX. Notice that U(Xx{x}), as |X|2. Thus, for any zU(Xx{x}),

    dU(u,z)=dU(v,z)=η(x)+dU(x,z)2η(x)dU(u,v),

    i.e., dU(u,z)dU(u,v) and dU(v,z)dU(u,v), and so zLMfM{u,v}, which is a contradiction. Therefore, this case does not occur.

    Case 3. uX and vXu. If Xu={v}, then (ⅰ) follows. Now, if |Xu|2, then for any vXu{v} we have dU(u,v)=dU(u,v)=η(u), and so v[u,v]={u,v}, v[u,v]={u,v} and vLMfM{u,v}, which implies that u[v,v]. As a result,

    dXu(v,v)min{2η(u),dXu(v,v)}=dU(v,v)=dU(v,u)+dU(u,v)=2η(u).

    Therefore, (ⅰ) follows.

    Case 4. uXx and vXx for two distinct points x,xX. Suppose that there exists uXx{u}. In this case,

    dU(u,v)=dU(u,v)=η(x)+dX(x,x)+η(x)>2η(x)dU(u,u),

    which contradicts that uLMfM{u,v}. Therefore, Xx={u}, and so (ⅰ) follows.

    Case 5. uX and vXx for some point xX{u}. Suppose that there exists vXx{v}. In this case,

    dU(u,v)=dU(u,v)=η(x)+dX(x,u)2η(x)dU(v,v),

    which contradicts that vLMfM{u,v}. Therefore, (ⅰ) follows.

    According to the five cases above, the proof is complete.

    The following result is a particular case of Theorem 6.2.

    Corollary 6.3. Let MfM be a corona metric space and X be the set of points of M. If 1D(Mx)<2η(x) for every xX, then the following statements are equivalent.

    (ⅰ) MfM has a universal line.

    (ⅱ) There exist two different points x,xX such that [x,x]={x,x} and the line LM{x,x} is universal.

    A metric space (Y,dY) is metrically convex, or convex in the sense of Menger, if for all different points u,vY there exists wY such that w[u,v]{u,v}, i.e., dY(u,v)=dY(u,w)+dY(w,v).

    It is not difficult to check that the following result follows from Theorem 6.2.

    Corollary 6.4. Let MfM be a corona metric space and X be the set of points of M. If |Xx|2, and the gravitational metric space Mx is metrically convex for every xX, then the following statements are equivalent.

    (ⅰ) MfM has a universal line.

    (ⅱ) There exist two different points x,xX such that [x,x]={x,x}, and the line LM{x,x} is universal.

    Proof. We only need to observe that if |Xx|2, and the gravitational metric space Mx is metrically convex for every xX, then condition (ⅰ) in Theorem 6.2 does not hold.

    In Section 4 we learned that the gravitational metric space Mx, associated to a metric space Mx, inherits several properties from Mx. For instance, Theorem 4.2 states that the completeness is one of these hereditary properties. It is not difficult to check that, if Mx has gravitational constant η(x)<D(Mx)2, then the convexity of Mx in the sense of Menger is not inherited by Mx, and neither is it inherited when Mx is a complete metric space. Even so, we will show how to obtain a result in the direction of Corollary 6.4, in which we impose the convexity on Mx instead of imposing it on Mx. However, in return we will have to impose the completeness on Mx. In short, to obtain the result we will not need Mx to inherit the convexity of Mx.

    One of our main tools will be the Menger theorem on complete and metrically convex metric spaces. First, let us recall that a subset S of points of a metric space (X,d) is called a metric segment between two distinct points x,yX if there exist a closed interval [a,b] on the real line and an isometry λ:[a,b]S such that λ([a,b])=S, λ(a)=x, and λ(b)=y.

    Theorem 6.5 (Menger). Each two distinct points of a complete and metrically convex metric space are joined by a metric segment of the space.

    The following result is obtained by combining Theorems 6.2 and 6.5.

    Theorem 6.6. Let MfM be a corona metric space and X be the set of points of M. If |Xx|2 and Mx is a complete and metrically convex metric space for every xX, then the following statements are equivalent.

    (ⅰ) MfM has a universal line.

    (ⅱ) There exist two different points x,xX such that [x,x]={x,x}, and the line LM{x,x} is universal.

    Proof. For any xX, assume that |Xx|2, and Mx is a complete and metrically convex metric space. By Theorem 6.2, we only need to prove that for any yXx there exists zXx{y} such that dXx(y,z)<2η(x).

    To this end, let y,yXx be two distinct points of Mx, and let t=dXx(y,y). By Theorem 6.5, there exists a metric segment Sy,yXx with endpoints y,y, i.e., there exists an isometry λ:[0,t]Sy,y such that λ(0)=y and λ(t)=y. Hence, for any 0<t<min{t,2η(x)}, there exists zSy,y{y,y} such that λ(t)=z, and, as a result, dXx(y,z)=t<2η(x). Therefore, the result follows.

    In this paper, we have introduced the study of corona metric spaces as a natural generalization of the theory of corona product graphs. In particular, the concept of a corona metric space was defined in such a way that the set of vertices of any corona product graph, equipped with the shortest path distance, is a corona metric space.

    An important role in this theory is played by the gravitational metric spaces, as many properties of the corona metric spaces are discussed on the basis of properties of the gravitational metric spaces. In particular, we have considered basic properties including completeness, boundedness, compactness and separability. Furthermore, we have obtained a formula for the metric dimension of a corona metric space, and we have derived some results linking the metric dimension with some properties discussed previously. Finally, we have obtained necessary and sufficient conditions for the existence of universal lines in corona metric spaces.

    The author declare that he has no conflict of interest.



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