# The Topology ofGB-Metric Spaces

Authors: Akbar Dehghan Nezhad; Zohreh Aral
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We define the concepts ofGB-metric in sets over σ-complete Boolean algebra and obtain some applications of them on the theory of topology. We also study some related properties of them.


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## Body

## 1. Introduction

Numerous studies have been made concerning geometries and topologies induced in sets by general distance functions. A formulation of the notion “generalized metric space (orG-metric space)" has been given [1]. In this paper, we begin the elaboration of the topology induced in sets over σ-complete Boolean algebra.In this paper,B shall always denote a σ-complete Boolean algebra. In B, we denote the operations of join, meet, and complement by a∨b, a∧b and a′, respectively.


## 2. TheGB-Metric Spaces

In 1952, a new structure of metric spaces, so calledB-metric space was introduced by Ellis and Sprinkle [2], on the set X to Boolean algebra (for details see [3, 4]).Definition 2.1 (see [2]).
AB-metric space is a set X with a map d:X×X→B (B is σ-Boolean algebra) with the properties(1)
d(ξ,η)=0 if and only if ξ=η,(2)
d(η,ξ)=d(ξ,η), (symmetry), and(3)
d(ξ,ζ)≤d(ξ,η)∨d(η,ζ), for all ξ, η, ζ belong to X.In [1], the present author has introduced a new structure of metric spaces which is a generalized idea of the ordinary metric space. The term generalized metric space is used in [5, 6].Definition 2.2 (see [7]).
LetX be a nonempty set and G:X×X×X→[0,∞) be a function satisfying the following properties:(G1)
G(x,y,z)=0 if x=y=z,(G2)
0<G(x,x,y); for all x,y∈X, with x≠y,(G3)
G(x,x,y)≤G(x,y,z), for all x,y,z∈X with  z≠y,(G4)
G(x,y,z)=G(x,z,y)=G(y,z,x)=⋯, (symmetry in all three variables), and(G5)
G(x,y,z)≤G(x,a,a)+G(a,y,z), for all x,y,z,a∈X, (rectangle inequality).


### 2.1. Topology OfGB-Metric Spaces

In this section we define generalizedB-metric (or GB-metric) space and introduce some basic notions and results that are used in sequel.Definition 2.3.
LetX be a non empty set and GB:X×X×X→B, be a function satisfying the following properties(GB1)
GB(x,y,z)=0 if x=y=z,(GB2)
0<GB(x,x,y);   for  all x,y∈X, with x≠y,(GB3)
GB(x,x,y)≤GB(x,y,z),  for  all x,y,z∈X  with  z≠y,(GB4)
GB(x,y,z)=GB(x,z,y)=GB(y,z,x)=⋯, (symmetry in all three variables), and(GB5)
GB(x,y,z)≤GB(x,a,a)∨GB(a,y,z),forall x,y,z,a∈X, (rectangle inequality).Then, the function GB is called a generalized B-metric, or, more specifically, a GB-metric on X, and the pair (X,GB) is called a GB-metric space.Example 2.4.
Put𝕎=ℕ∪{0}. Define a map GB:𝕎×𝕎×𝕎→{0,1} by
(2.1)GB(m,n,k)={0m=n=k,1else.
The mapGB is a GB-metric on  𝕎.Remark 2.5.
We can show that aGB-metric space is a generalized B-metric space over X.Proposition 2.6.
EveryGB-metric space (X,GB) will define a B-metric (X,dGB) by dGB(x,y)=GB(x,y,y).Proof.
Conditions 1 and 2 ofB-metric are clearly and 3 follows from (GB5).Proposition 2.7.
LetGB be GB-metric on a ring X. For an element a∈X, the following maps are GB-metrics on the ring X:(1)
GB*(x,y,z)=GB(a-x,a-y,a-z) and(2)
GB⋆(x,y,z)=GB(x,y,z)∨GB(a-x,a-y,a-z).Proposition 2.8.
Let(X,d) be a B-metric space. Define a function, GB:X×X×X→B by GB(x,y,z)=d(x,y)∨d(y,z)∨d(z,x) for x,y,z∈X. The map GB is a GB-metric on X, and consequently (X,GB) is a GB-metric space.Proof.
Conditions(GB1),(GB2), and (GB4) are clear. We show that (GB3) and (GB5) are valid too. 									(2.2)GB(x,x,y)=d(x,x)∨d(x,y)∨d(x,y)<d(x,z)∨d(z,y)<d(x,z)∨d(z,y)∨d(x,y)=GB(x,y,z).
And for (GB5), we have
(2.3)GB(x,y,z)=d(x,y)∨d(y,z)∨d(z,x)<d(x,a)∨d(y,a)∨d(z,a)∨d(y,z)=d(x,a)∨GB(a,y,z)=GB(a,a,x)∨GB(a,y,z).Proposition 2.9.
Let(X,GB) be a GB-metric space, then for all x,y,z,a∈X, it follows that:(1)
GB(x,y,z)≤GB(x,x,y)∨GB(x,x,z),(2)
GB(x,y,y)=GB(x,x,y),(3)
GB(x,y,z)≤GB(x,a,z)∨GB(a,y,z), and(4)
GB(x,y,z)≤GB(x,a,a)∨GB(y,a,a)∨GB(z,a,a).Proposition 2.10.
Let(X,GB) be a GB-metric space, then the following are equivalent:(1)
GB(x,y,y)≤GB(x,y,a) for all x,y,z,a∈X,(2)
GB(x,y,z)≤GB(x,y,a)∨GB(z,y,b) for all x,y,z,a∈X.Proof.
Use (2) of Proposition2.9.Definition 2.11.
Let(X,GB) be a GB-metric. For all x0∈X, r∈B∖{0}, the GB-ball with center x0 and radius r is BGB(x0,r)={y∈X∣GB(x0,y,y)<r}. For any GB-ball, we can define BGB¯(x0,r)={y∈X∣GB(x0,y,y)≤r}.Proposition 2.12.
Let(X,GB) be a G-metric space. Then for all x0∈X and r∈B∖{0}, we have(1)
IfGB(x0,x,y)<r, then x,y∈BGB(x0,r) and(2)
ify∈BGB(x0,r), then there exists one element δ∈B∖{0} such that, BGB(y,δ)⊆BGB(x0,r).Proof.
(1) follows directlyGB(x,x,y)≤GB(x,y,z). Put δ=r-GB(x,y,y) and use (GB5), then deduce (2).It follows from (2) of the above proposition that the family of allGB-balls (2.4)β={BGB(x,r)∣x∈X,r∈B∖{0}}
is the base of topology τGB on X, we call the GB-metric topology.Definition 2.13.
Let(X,GB) be a GB-metric space. The sequence {xn}⊆X is GB-convergent to x if it converges to x in the GB-metric topology, τGB.Remark 2.14.
Another topological notions asGB-Hausdorff, GB-compact, GB-normal, and … define similarly as usual.Definition 2.15.
Let(X,GB) and (X′,GB′) be GB-metric spaces, a function f:X→X′ is GB-continuous at a point x0∈X if f-1(BGB′(f(x0),r))∈τ(G) for all r∈B∖{0}. We say f is GB-continuous if it is GB-continuous at all points of X, that is, continuous as a function from X with the τGB-topology to X′ with the τGB′-topology.Theorem 2.16.
Let(X,GB) be a GB-metric space and f be a self map of X into itself. Suppose f is GB-continuous at x0∈X, if there is a point x∈X such that the sequence of iterates {fn(x)} converges to x0, then f(x0)=x0.Proof.
From(GB5), we derive GB(f(x0),x0,x0)≤GB(f(x0),fn(x),fn(x))∨GB(fn(x),x0,x0)=GB(f(x0),f(fn-1(x)),f(fn-1(x)))∨GB(fn(x),x0,x0) as n→∞we deduce, GB(f(x0),f(fn-1(x)),f(fn-1(x)))∨GB(fn(x),x0,x0)→0, so the above equation will be GB(f(x0),x0,x0)≤0, therefore, f(x0)=x0.Theorem 2.17.
Let(X,GB) be a GB-metric space and f:X→X be GB-continuous. If for x∈X, the sequence {fn(x)} has a convergent subsequence {fni(x)}GB-converges to p, and GB(fni(x),fni+1(x),fni+1(x))→0, then f has a fixpoint.Proof.
Since{fni(x)}→p, then by GB-continuity of f we have f(fni(x))=fni+1(x)→f(p), and we have
(2.5)GB(p,f(p),f(p))≤GB(p,fni(x),fni(x))∨GB(fni(x),fni+1(x),fni+1(x))∨GB(fni+1(x),f(p),f(p)).
As i→∞ we get GB(p,f(p),f(p))≤0, which implies that f(p)=p.Theorem 2.18.
AGB-metric space X is GB-Hausdorff.Proof.
Consider two elementsx, y belong to X that x≠y, and two subsets A={x∈X∣GB(x,x,z)<GB(x,x,y)} and B={x∈X∣GB(x,x,y)<GB(x,x,z)}. These are two disjoint neighbourhoods of x and y, respectively. It is sufficient to prove that the subsets A and B are open. Suppose t∈A. Put GB(t,t,z)=r1, GB(t,t,y)=r2 and δ=r2-r1. If s∈BGB(t,δ), then GB(s,s,z)<GB(t,t,z)∨GB(s,s,t)<r2=GB(t,t,y)<GB(t,t,s)∨GB(s,s,y)=GB(s,s,y), therefore, s∈A and BGB(t,δ)⊂A. This proves X is Hausdorff.Theorem 2.19.
AGB-metric space X is GB-normal.Proof.
LetGB(x,x,D)=⋀{GB(x,x,y)∣y∈D}. Consider two closed disjoint subsets E and F of X. Put A={x∈X∣(x,x,F)<GB(x,x,E)} and B={x∈XGB(x,x,E)<GB(x,x,F)}. It is clear that A,B are disjoint open subsets of X, E⊂A, and F⊂B.Corollary 2.20.
AGB-metric space X is GB-regular.


### 2.2. Monoid InvariantGB-Metric

Definition 2.21.
AGB-metric gb on a (X,⊙) is m-invariant (or monoid invariant) when GB(x⊙a,y⊙a,z⊙a)≤GB(x,y,z) and GB(a⊙x,a⊙y,a⊙z)≤GB(x,y,z) are valid for any x,y,z,a∈X.Proposition 2.22.
Let(X,⊙) be a monoid. If GB is an m-invariant GB-metric on X, then(1)
GB(x⊙a,y⊙b,z⊙c)≤GB(x,y,z)∨GB(a,b,c) and(2)
GB(x1⊙x2⊙x3⊙⋯⊙xn,y1⊙y2⊙y3⊙⋯⊙yn,z1⊙z2⊙z3⊙⋯⊙zn)≤GB(x1,y1,z1)∨GB(x2,y2,z2)∨GB(x3,y3,z3)∨⋯∨GB(xn,yn,zn).Proof.
(1) We have(2.6)gb(x⊙a,y⊙b,z⊙c)≤gb(x⊙a,y⊙a,y⊙a)∨gb(y⊙a,y⊙b,z⊙c)≤gb(x,y,y)∨gb(y⊙a,y⊙b,y⊙c)∨gb(y⊙c,y⊙c,z⊙c)≤gb(x,y,y)∨gb(a,b,c)∨gb(y,y,z)≤gb(x,y,z)∨gb(x,y,z)∨gb(a,b,c)=gb(x,y,z)∨gb(a,b,c).
(2) It concludes by induction.Put(X,⊙) is a monoid and GB be a GB-metric on X. We call GB is left effective on X if GB(a⊙x,a⊙y,a⊙z)=GB(x,y,z) for all x,y,z∈X then a=e. And we say GB is left free GB-metric on X if for a triple (x,y,z)∈X×X×X, GB(a⊙x,a⊙y,a⊙z)=GB(x,y,z) then a=e. The right effective (free) is defined samely, and we call GB is effective (free) if it is left and right effective (free) on X. Immediately we deduce, if GB is free GB-metric on (X,⊙) then it is effective GB-metric.Proposition 2.23.
Putf:(X,⊙)→(X′,⊗) is a monoid homomorphism and GB′ is a GB-metric on X′ the following results are valid:(1)
GB:X×X×X→B, (x,y,z)↦GB′(f(x),f(y),f(z)) is a GB-metric on (X,⊙).(2)
IfGB′ is m-invariant, then GB is m-invariant.(3)
TheGB-metric GB on X is free (or effective) if GB′ is free (or effective) on X′.LetE and F be subsets of X. The set E is congruent to F, written E≈F, provided there exist a map f:E→F that for any x,y,z∈E, GB(x,y,z)=GB(f(x),f(y),f(z)) and that F is congruent to E by the map f-1:F→E. The map f(orf-1) is called a congruence between E and F. Clearly every single set {x}, x∈X and each map f, {x} congruent to {f(x)}.Definition 2.24.
A motion ofX is a congruence of X with itself.Proposition 2.25.
Let(X,GB) be a GB-metric space. The set ℳ(X)={f:X→X∣fis motion} is a group.Proof.
The identity map belongs toℳ(X) so ℳ(X) is not empty. Group action is composition of maps, so the identity map is the identity element. If g∈ℳ(X), then GB(g-1(x),g-1(y),g-1(z))=GB(g(g-1(x)),g(g-1(y)),g(g-1(z)))=(x,y,z), so g-1∈ℳ(x).Theorem 2.26.
Let(X,GB) be a GB-metric space and f:X→X is a motion of X, with the properties: if {fn}→x, then GB(fni+1(x),fni+1(x),fn(x))→0, then the set of fixpoints of f (or F(f)) is equal to the set of fixpoints of fn(orF(fn)) for all n∈ℕ.Proof.
ClearlyF(f)⊆F(fn). Suppose x∈F(fn), so the sequence {fnk(x)}k=1∞→x, because fkn(x)=x, for all k∈ℕ. Now by Theorem 2.16, we conclude f(x)=x.Proposition 2.27.
Let(X,*) be a monoid and GB be a GB-metric on X.(1)
If for alla∈X, La and Ra be motions of X, then GB is m-invariant.(2)
La (or Ra) (a≠e) cannot be a motion of X, if GB is effective.Example 2.28.
TheGB-metric defined in Example 2.4 is invariant on the monoid (𝕎,+), but it is not effective. Clearly for all k∈ℕ, fk:𝕎→𝕎, n↦n+k is a motion.


## 2.1. Topology OfGB-Metric Spaces

In this section we define generalizedB-metric (or GB-metric) space and introduce some basic notions and results that are used in sequel.Definition 2.3.
LetX be a non empty set and GB:X×X×X→B, be a function satisfying the following properties(GB1)
GB(x,y,z)=0 if x=y=z,(GB2)
0<GB(x,x,y);   for  all x,y∈X, with x≠y,(GB3)
GB(x,x,y)≤GB(x,y,z),  for  all x,y,z∈X  with  z≠y,(GB4)
GB(x,y,z)=GB(x,z,y)=GB(y,z,x)=⋯, (symmetry in all three variables), and(GB5)
GB(x,y,z)≤GB(x,a,a)∨GB(a,y,z),forall x,y,z,a∈X, (rectangle inequality).Then, the function GB is called a generalized B-metric, or, more specifically, a GB-metric on X, and the pair (X,GB) is called a GB-metric space.Example 2.4.
Put𝕎=ℕ∪{0}. Define a map GB:𝕎×𝕎×𝕎→{0,1} by
(2.1)GB(m,n,k)={0m=n=k,1else.
The mapGB is a GB-metric on  𝕎.Remark 2.5.
We can show that aGB-metric space is a generalized B-metric space over X.Proposition 2.6.
EveryGB-metric space (X,GB) will define a B-metric (X,dGB) by dGB(x,y)=GB(x,y,y).Proof.
Conditions 1 and 2 ofB-metric are clearly and 3 follows from (GB5).Proposition 2.7.
LetGB be GB-metric on a ring X. For an element a∈X, the following maps are GB-metrics on the ring X:(1)
GB*(x,y,z)=GB(a-x,a-y,a-z) and(2)
GB⋆(x,y,z)=GB(x,y,z)∨GB(a-x,a-y,a-z).Proposition 2.8.
Let(X,d) be a B-metric space. Define a function, GB:X×X×X→B by GB(x,y,z)=d(x,y)∨d(y,z)∨d(z,x) for x,y,z∈X. The map GB is a GB-metric on X, and consequently (X,GB) is a GB-metric space.Proof.
Conditions(GB1),(GB2), and (GB4) are clear. We show that (GB3) and (GB5) are valid too. 									(2.2)GB(x,x,y)=d(x,x)∨d(x,y)∨d(x,y)<d(x,z)∨d(z,y)<d(x,z)∨d(z,y)∨d(x,y)=GB(x,y,z).
And for (GB5), we have
(2.3)GB(x,y,z)=d(x,y)∨d(y,z)∨d(z,x)<d(x,a)∨d(y,a)∨d(z,a)∨d(y,z)=d(x,a)∨GB(a,y,z)=GB(a,a,x)∨GB(a,y,z).Proposition 2.9.
Let(X,GB) be a GB-metric space, then for all x,y,z,a∈X, it follows that:(1)
GB(x,y,z)≤GB(x,x,y)∨GB(x,x,z),(2)
GB(x,y,y)=GB(x,x,y),(3)
GB(x,y,z)≤GB(x,a,z)∨GB(a,y,z), and(4)
GB(x,y,z)≤GB(x,a,a)∨GB(y,a,a)∨GB(z,a,a).Proposition 2.10.
Let(X,GB) be a GB-metric space, then the following are equivalent:(1)
GB(x,y,y)≤GB(x,y,a) for all x,y,z,a∈X,(2)
GB(x,y,z)≤GB(x,y,a)∨GB(z,y,b) for all x,y,z,a∈X.Proof.
Use (2) of Proposition2.9.Definition 2.11.
Let(X,GB) be a GB-metric. For all x0∈X, r∈B∖{0}, the GB-ball with center x0 and radius r is BGB(x0,r)={y∈X∣GB(x0,y,y)<r}. For any GB-ball, we can define BGB¯(x0,r)={y∈X∣GB(x0,y,y)≤r}.Proposition 2.12.
Let(X,GB) be a G-metric space. Then for all x0∈X and r∈B∖{0}, we have(1)
IfGB(x0,x,y)<r, then x,y∈BGB(x0,r) and(2)
ify∈BGB(x0,r), then there exists one element δ∈B∖{0} such that, BGB(y,δ)⊆BGB(x0,r).Proof.
(1) follows directlyGB(x,x,y)≤GB(x,y,z). Put δ=r-GB(x,y,y) and use (GB5), then deduce (2).It follows from (2) of the above proposition that the family of allGB-balls (2.4)β={BGB(x,r)∣x∈X,r∈B∖{0}}
is the base of topology τGB on X, we call the GB-metric topology.Definition 2.13.
Let(X,GB) be a GB-metric space. The sequence {xn}⊆X is GB-convergent to x if it converges to x in the GB-metric topology, τGB.Remark 2.14.
Another topological notions asGB-Hausdorff, GB-compact, GB-normal, and … define similarly as usual.Definition 2.15.
Let(X,GB) and (X′,GB′) be GB-metric spaces, a function f:X→X′ is GB-continuous at a point x0∈X if f-1(BGB′(f(x0),r))∈τ(G) for all r∈B∖{0}. We say f is GB-continuous if it is GB-continuous at all points of X, that is, continuous as a function from X with the τGB-topology to X′ with the τGB′-topology.Theorem 2.16.
Let(X,GB) be a GB-metric space and f be a self map of X into itself. Suppose f is GB-continuous at x0∈X, if there is a point x∈X such that the sequence of iterates {fn(x)} converges to x0, then f(x0)=x0.Proof.
From(GB5), we derive GB(f(x0),x0,x0)≤GB(f(x0),fn(x),fn(x))∨GB(fn(x),x0,x0)=GB(f(x0),f(fn-1(x)),f(fn-1(x)))∨GB(fn(x),x0,x0) as n→∞we deduce, GB(f(x0),f(fn-1(x)),f(fn-1(x)))∨GB(fn(x),x0,x0)→0, so the above equation will be GB(f(x0),x0,x0)≤0, therefore, f(x0)=x0.Theorem 2.17.
Let(X,GB) be a GB-metric space and f:X→X be GB-continuous. If for x∈X, the sequence {fn(x)} has a convergent subsequence {fni(x)}GB-converges to p, and GB(fni(x),fni+1(x),fni+1(x))→0, then f has a fixpoint.Proof.
Since{fni(x)}→p, then by GB-continuity of f we have f(fni(x))=fni+1(x)→f(p), and we have
(2.5)GB(p,f(p),f(p))≤GB(p,fni(x),fni(x))∨GB(fni(x),fni+1(x),fni+1(x))∨GB(fni+1(x),f(p),f(p)).
As i→∞ we get GB(p,f(p),f(p))≤0, which implies that f(p)=p.Theorem 2.18.
AGB-metric space X is GB-Hausdorff.Proof.
Consider two elementsx, y belong to X that x≠y, and two subsets A={x∈X∣GB(x,x,z)<GB(x,x,y)} and B={x∈X∣GB(x,x,y)<GB(x,x,z)}. These are two disjoint neighbourhoods of x and y, respectively. It is sufficient to prove that the subsets A and B are open. Suppose t∈A. Put GB(t,t,z)=r1, GB(t,t,y)=r2 and δ=r2-r1. If s∈BGB(t,δ), then GB(s,s,z)<GB(t,t,z)∨GB(s,s,t)<r2=GB(t,t,y)<GB(t,t,s)∨GB(s,s,y)=GB(s,s,y), therefore, s∈A and BGB(t,δ)⊂A. This proves X is Hausdorff.Theorem 2.19.
AGB-metric space X is GB-normal.Proof.
LetGB(x,x,D)=⋀{GB(x,x,y)∣y∈D}. Consider two closed disjoint subsets E and F of X. Put A={x∈X∣(x,x,F)<GB(x,x,E)} and B={x∈XGB(x,x,E)<GB(x,x,F)}. It is clear that A,B are disjoint open subsets of X, E⊂A, and F⊂B.Corollary 2.20.
AGB-metric space X is GB-regular.


## 2.2. Monoid InvariantGB-Metric

Definition 2.21.
AGB-metric gb on a (X,⊙) is m-invariant (or monoid invariant) when GB(x⊙a,y⊙a,z⊙a)≤GB(x,y,z) and GB(a⊙x,a⊙y,a⊙z)≤GB(x,y,z) are valid for any x,y,z,a∈X.Proposition 2.22.
Let(X,⊙) be a monoid. If GB is an m-invariant GB-metric on X, then(1)
GB(x⊙a,y⊙b,z⊙c)≤GB(x,y,z)∨GB(a,b,c) and(2)
GB(x1⊙x2⊙x3⊙⋯⊙xn,y1⊙y2⊙y3⊙⋯⊙yn,z1⊙z2⊙z3⊙⋯⊙zn)≤GB(x1,y1,z1)∨GB(x2,y2,z2)∨GB(x3,y3,z3)∨⋯∨GB(xn,yn,zn).Proof.
(1) We have(2.6)gb(x⊙a,y⊙b,z⊙c)≤gb(x⊙a,y⊙a,y⊙a)∨gb(y⊙a,y⊙b,z⊙c)≤gb(x,y,y)∨gb(y⊙a,y⊙b,y⊙c)∨gb(y⊙c,y⊙c,z⊙c)≤gb(x,y,y)∨gb(a,b,c)∨gb(y,y,z)≤gb(x,y,z)∨gb(x,y,z)∨gb(a,b,c)=gb(x,y,z)∨gb(a,b,c).
(2) It concludes by induction.Put(X,⊙) is a monoid and GB be a GB-metric on X. We call GB is left effective on X if GB(a⊙x,a⊙y,a⊙z)=GB(x,y,z) for all x,y,z∈X then a=e. And we say GB is left free GB-metric on X if for a triple (x,y,z)∈X×X×X, GB(a⊙x,a⊙y,a⊙z)=GB(x,y,z) then a=e. The right effective (free) is defined samely, and we call GB is effective (free) if it is left and right effective (free) on X. Immediately we deduce, if GB is free GB-metric on (X,⊙) then it is effective GB-metric.Proposition 2.23.
Putf:(X,⊙)→(X′,⊗) is a monoid homomorphism and GB′ is a GB-metric on X′ the following results are valid:(1)
GB:X×X×X→B, (x,y,z)↦GB′(f(x),f(y),f(z)) is a GB-metric on (X,⊙).(2)
IfGB′ is m-invariant, then GB is m-invariant.(3)
TheGB-metric GB on X is free (or effective) if GB′ is free (or effective) on X′.LetE and F be subsets of X. The set E is congruent to F, written E≈F, provided there exist a map f:E→F that for any x,y,z∈E, GB(x,y,z)=GB(f(x),f(y),f(z)) and that F is congruent to E by the map f-1:F→E. The map f(orf-1) is called a congruence between E and F. Clearly every single set {x}, x∈X and each map f, {x} congruent to {f(x)}.Definition 2.24.
A motion ofX is a congruence of X with itself.Proposition 2.25.
Let(X,GB) be a GB-metric space. The set ℳ(X)={f:X→X∣fis motion} is a group.Proof.
The identity map belongs toℳ(X) so ℳ(X) is not empty. Group action is composition of maps, so the identity map is the identity element. If g∈ℳ(X), then GB(g-1(x),g-1(y),g-1(z))=GB(g(g-1(x)),g(g-1(y)),g(g-1(z)))=(x,y,z), so g-1∈ℳ(x).Theorem 2.26.
Let(X,GB) be a GB-metric space and f:X→X is a motion of X, with the properties: if {fn}→x, then GB(fni+1(x),fni+1(x),fn(x))→0, then the set of fixpoints of f (or F(f)) is equal to the set of fixpoints of fn(orF(fn)) for all n∈ℕ.Proof.
ClearlyF(f)⊆F(fn). Suppose x∈F(fn), so the sequence {fnk(x)}k=1∞→x, because fkn(x)=x, for all k∈ℕ. Now by Theorem 2.16, we conclude f(x)=x.Proposition 2.27.
Let(X,*) be a monoid and GB be a GB-metric on X.(1)
If for alla∈X, La and Ra be motions of X, then GB is m-invariant.(2)
La (or Ra) (a≠e) cannot be a motion of X, if GB is effective.Example 2.28.
TheGB-metric defined in Example 2.4 is invariant on the monoid (𝕎,+), but it is not effective. Clearly for all k∈ℕ, fk:𝕎→𝕎, n↦n+k is a motion.



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*Source: 523453-2011-06-23.xml*