Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 523453, 6 pages
http://dx.doi.org/10.5402/2011/523453
Research Article

The Topology of 𝐺𝐡-Metric Spaces

Faculty of Mathematics, Yazd University, Yazd, Iran

Received 24 February 2011; Accepted 27 April 2011

Academic Editor: M.Β Azaiez

Copyright Β© 2011 Akbar Dehghan Nezhad and Zohreh Aral. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define the concepts of 𝐺𝐡-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.

1. Introduction

Numerous studies have been made concerning geometries and topologies induced in sets by general distance functions. A formulation of the notion β€œgeneralized metric space (or 𝐺-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, 𝐡 shall always denote a 𝜎-complete Boolean algebra. In 𝐡, we denote the operations of join, meet, and complement by π‘Žβˆ¨π‘, π‘Žβˆ§π‘ and π‘Žβ€², respectively.

2. The 𝐺𝐡-Metric Spaces

In 1952, a new structure of metric spaces, so called 𝐡-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]). A 𝐡-metric space is a set 𝑋 with a map π‘‘βˆΆπ‘‹Γ—π‘‹β†’π΅ (𝐡 is 𝜎-Boolean algebra) with the properties(1)𝑑(πœ‰,πœ‚)=0 if and only if πœ‰=πœ‚,(2)𝑑(πœ‚,πœ‰)=𝑑(πœ‰,πœ‚), (symmetry), and (3)𝑑(πœ‰,𝜁)≀𝑑(πœ‰,πœ‚)βˆ¨π‘‘(πœ‚,𝜁), for all πœ‰, πœ‚, 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]). Let 𝑋 be a nonempty set and πΊβˆΆπ‘‹Γ—π‘‹Γ—π‘‹β†’[0,∞) be a function satisfying the following properties:(𝐺1)𝐺(π‘₯,𝑦,𝑧)=0 if π‘₯=𝑦=𝑧,(𝐺2)0<𝐺(π‘₯,π‘₯,𝑦); for all π‘₯,π‘¦βˆˆπ‘‹, with π‘₯≠𝑦,(𝐺3)𝐺(π‘₯,π‘₯,𝑦)≀𝐺(π‘₯,𝑦,𝑧), for all π‘₯,𝑦,π‘§βˆˆπ‘‹ with 𝑧≠𝑦,(𝐺4)G(π‘₯,𝑦,𝑧)=𝐺(π‘₯,𝑧,𝑦)=𝐺(𝑦,𝑧,π‘₯)=β‹―, (symmetry in all three variables), and(𝐺5)𝐺(π‘₯,𝑦,𝑧)≀𝐺(π‘₯,π‘Ž,π‘Ž)+𝐺(π‘Ž,𝑦,𝑧), for all π‘₯,𝑦,𝑧,π‘Žβˆˆπ‘‹, (rectangle inequality).

2.1. Topology Of 𝐺𝐡-Metric Spaces

In this section we define generalized 𝐡-metric (or 𝐺𝐡-metric) space and introduce some basic notions and results that are used in sequel.

Definition 2.3. Let 𝑋 be a non empty set and πΊπ΅βˆΆπ‘‹Γ—π‘‹Γ—π‘‹β†’π΅, be a function satisfying the following properties(𝐺𝐡1)𝐺𝐡(π‘₯,𝑦,𝑧)=0 if π‘₯=𝑦=𝑧,(𝐺𝐡2)0<𝐺𝐡(π‘₯,π‘₯,𝑦);   for  all π‘₯,π‘¦βˆˆπ‘‹, with π‘₯≠𝑦,(𝐺𝐡3)𝐺𝐡(π‘₯,π‘₯,𝑦)≀𝐺𝐡(π‘₯,𝑦,𝑧),  for  all π‘₯,𝑦,π‘§βˆˆπ‘‹ with 𝑧≠𝑦,(𝐺𝐡4)𝐺𝐡(π‘₯,𝑦,𝑧)=𝐺𝐡(π‘₯,𝑧,𝑦)=𝐺𝐡(𝑦,𝑧,π‘₯)=β‹―, (symmetry in all three variables), and(𝐺𝐡5)𝐺𝐡(π‘₯,𝑦,𝑧)≀𝐺𝐡(π‘₯,π‘Ž,π‘Ž)∨𝐺𝐡(π‘Ž,𝑦,𝑧),forall π‘₯,𝑦,𝑧,π‘Žβˆˆπ‘‹, (rectangle inequality).Then, the function 𝐺𝐡 is called a generalized 𝐡-metric, or, more specifically, a 𝐺𝐡-metric on 𝑋, and the pair (𝑋,𝐺𝐡) is called a 𝐺𝐡-metric space.

Example 2.4. Put π•Ž=β„•βˆͺ{0}. Define a map πΊπ΅βˆΆπ•ŽΓ—π•ŽΓ—π•Žβ†’{0,1} by 𝐺𝐡1(π‘š,𝑛,π‘˜)=0π‘š=𝑛=π‘˜,else.(2.1)
The map 𝐺𝐡 is a 𝐺𝐡-metric onβ€‰β€‰π•Ž.

Remark 2.5. We can show that a 𝐺𝐡-metric space is a generalized 𝐡-metric space over 𝑋.

Proposition 2.6. Every 𝐺𝐡-metric space (𝑋,𝐺𝐡) will define a 𝐡-metric (𝑋,𝑑𝐺𝐡) by 𝑑𝐺𝐡(π‘₯,𝑦)=𝐺𝐡(π‘₯,𝑦,𝑦).

Proof. Conditions 1 and 2 of 𝐡-metric are clearly and 3 follows from (𝐺𝐡5).

Proposition 2.7. Let 𝐺𝐡 be 𝐺𝐡-metric on a ring 𝑋. For an element π‘Žβˆˆπ‘‹, the following maps are 𝐺𝐡-metrics on the ring 𝑋: (1)πΊβˆ—π΅(π‘₯,𝑦,𝑧)=𝐺𝐡(π‘Žβˆ’π‘₯,π‘Žβˆ’π‘¦,π‘Žβˆ’π‘§) and(2)𝐺⋆𝐡(π‘₯,𝑦,𝑧)=𝐺𝐡(π‘₯,𝑦,𝑧)∨𝐺𝐡(π‘Žβˆ’π‘₯,π‘Žβˆ’π‘¦,π‘Žβˆ’π‘§).

Proposition 2.8. Let (𝑋,𝑑) be a 𝐡-metric space. Define a function, πΊπ΅βˆΆπ‘‹Γ—π‘‹Γ—X→𝐡 by 𝐺𝐡(π‘₯,𝑦,𝑧)=𝑑(π‘₯,𝑦)βˆ¨π‘‘(𝑦,𝑧)βˆ¨π‘‘(𝑧,π‘₯) for π‘₯,𝑦,π‘§βˆˆπ‘‹. The map 𝐺𝐡 is a 𝐺𝐡-metric on 𝑋, and consequently (𝑋,𝐺𝐡) is a 𝐺𝐡-metric space.

Proof. Conditions (𝐺𝐡1),(𝐺𝐡2), and (𝐺𝐡4) are clear. We show that (𝐺𝐡3) and (𝐺𝐡5) are valid too. 𝐺𝐡(π‘₯,π‘₯,𝑦)=𝑑(π‘₯,π‘₯)βˆ¨π‘‘(π‘₯,𝑦)βˆ¨π‘‘(π‘₯,𝑦)<𝑑(π‘₯,𝑧)βˆ¨π‘‘(𝑧,𝑦)<𝑑(π‘₯,𝑧)βˆ¨π‘‘(𝑧,𝑦)βˆ¨π‘‘(π‘₯,𝑦)=𝐺𝐡(π‘₯,𝑦,𝑧).(2.2) And for (𝐺𝐡5), we have 𝐺𝐡(π‘₯,𝑦,𝑧)=𝑑(π‘₯,𝑦)βˆ¨π‘‘(𝑦,𝑧)βˆ¨π‘‘(𝑧,π‘₯)<𝑑(π‘₯,π‘Ž)βˆ¨π‘‘(𝑦,π‘Ž)βˆ¨π‘‘(𝑧,π‘Ž)βˆ¨π‘‘(𝑦,𝑧)=𝑑(π‘₯,π‘Ž)∨𝐺𝐡(π‘Ž,𝑦,𝑧)=𝐺𝐡(π‘Ž,π‘Ž,π‘₯)∨𝐺𝐡(π‘Ž,𝑦,𝑧).(2.3)

Proposition 2.9. Let (𝑋,𝐺𝐡) be a 𝐺𝐡-metric space, then for all π‘₯,𝑦,𝑧,π‘Žβˆˆπ‘‹, it follows that: (1)𝐺𝐡(π‘₯,𝑦,𝑧)≀𝐺𝐡(π‘₯,π‘₯,𝑦)∨𝐺𝐡(π‘₯,π‘₯,𝑧),(2)𝐺𝐡(π‘₯,𝑦,𝑦)=𝐺𝐡(π‘₯,π‘₯,𝑦),(3)𝐺𝐡(π‘₯,𝑦,𝑧)≀𝐺𝐡(π‘₯,π‘Ž,𝑧)∨𝐺𝐡(π‘Ž,𝑦,𝑧), and(4)𝐺𝐡(π‘₯,𝑦,𝑧)≀𝐺𝐡(π‘₯,π‘Ž,π‘Ž)∨𝐺𝐡(𝑦,π‘Ž,π‘Ž)∨𝐺𝐡(𝑧,π‘Ž,π‘Ž).

Proposition 2.10. Let (𝑋,𝐺𝐡) be a 𝐺𝐡-metric space, then the following are equivalent: (1)𝐺𝐡(π‘₯,𝑦,𝑦)≀𝐺𝐡(π‘₯,𝑦,π‘Ž) for all π‘₯,𝑦,𝑧,π‘Žβˆˆπ‘‹,(2)𝐺𝐡(π‘₯,𝑦,𝑧)≀𝐺𝐡(π‘₯,𝑦,π‘Ž)∨𝐺𝐡(𝑧,𝑦,𝑏) for all π‘₯,𝑦,𝑧,π‘Žβˆˆπ‘‹.

Proof. Use (2) of Proposition 2.9.

Definition 2.11. Let (𝑋,𝐺𝐡) be a 𝐺𝐡-metric. For all π‘₯0βˆˆπ‘‹, π‘Ÿβˆˆπ΅β§΅{0}, the 𝐺𝐡-ball with center π‘₯0 and radius π‘Ÿ is 𝐡𝐺𝐡(π‘₯0,π‘Ÿ)={π‘¦βˆˆπ‘‹βˆ£πΊπ΅(π‘₯0,𝑦,𝑦)<π‘Ÿ}. For any 𝐺𝐡-ball, we can define 𝐡𝐺𝐡(π‘₯0,π‘Ÿ)={π‘¦βˆˆπ‘‹βˆ£πΊπ΅(π‘₯0,𝑦,𝑦)β‰€π‘Ÿ}.

Proposition 2.12. Let (𝑋,𝐺𝐡) be a 𝐺-metric space. Then for all π‘₯0βˆˆπ‘‹ and π‘Ÿβˆˆπ΅β§΅{0}, we have (1)If 𝐺𝐡(π‘₯0,π‘₯,𝑦)<π‘Ÿ, then π‘₯,π‘¦βˆˆπ΅πΊπ΅(π‘₯0,π‘Ÿ) and(2)if π‘¦βˆˆπ΅πΊπ΅(π‘₯0,π‘Ÿ), then there exists one element π›Ώβˆˆπ΅β§΅{0} such that, 𝐡𝐺𝐡(𝑦,𝛿)βŠ†π΅πΊπ΅(π‘₯0,π‘Ÿ).

Proof. (1) follows directly 𝐺𝐡(π‘₯,π‘₯,𝑦)≀𝐺𝐡(π‘₯,𝑦,𝑧). Put 𝛿=π‘Ÿβˆ’πΊπ΅(π‘₯,𝑦,𝑦) and use (𝐺𝐡5), then deduce (2).

It follows from (2) of the above proposition that the family of all 𝐺𝐡-balls 𝐡𝛽=𝐺𝐡(π‘₯,π‘Ÿ)∣π‘₯βˆˆπ‘‹,π‘Ÿβˆˆπ΅β§΅{0}(2.4) is the base of topology 𝜏𝐺𝐡 on 𝑋, we call the 𝐺𝐡-metric topology.

Definition 2.13. Let (𝑋,𝐺𝐡) be a 𝐺𝐡-metric space. The sequence {π‘₯𝑛}βŠ†π‘‹ is 𝐺𝐡-convergent to π‘₯ if it converges to π‘₯ in the 𝐺𝐡-metric topology, 𝜏𝐺𝐡.

Remark 2.14. Another topological notions as 𝐺𝐡-Hausdorff, 𝐺𝐡-compact, 𝐺𝐡-normal, and … define similarly as usual.

Definition 2.15. Let (𝑋,𝐺𝐡) and (π‘‹ξ…ž,πΊξ…žπ΅) be 𝐺𝐡-metric spaces, a function π‘“βˆΆπ‘‹β†’π‘‹ξ…ž is 𝐺𝐡-continuous at a point π‘₯0βˆˆπ‘‹ if π‘“βˆ’1(𝐡𝐺𝐡′(𝑓(π‘₯0),π‘Ÿ))∈𝜏(𝐺) for all π‘Ÿβˆˆπ΅β§΅{0}. We say 𝑓 is 𝐺𝐡-continuous if it is 𝐺𝐡-continuous at all points of 𝑋, that is, continuous as a function from 𝑋 with the 𝜏𝐺𝐡-topology to π‘‹ξ…ž with the πœξ…žπΊπ΅-topology.

Theorem 2.16. Let (𝑋,𝐺𝐡) be a 𝐺𝐡-metric space and 𝑓 be a self map of 𝑋 into itself. Suppose 𝑓 is 𝐺𝐡-continuous at π‘₯0βˆˆπ‘‹, if there is a point π‘₯βˆˆπ‘‹ such that the sequence of iterates {𝑓𝑛(π‘₯)} converges to π‘₯0, then 𝑓(π‘₯0)=π‘₯0.

Proof. From (𝐺𝐡5), we derive 𝐺𝐡(𝑓(π‘₯0),π‘₯0,π‘₯0)≀𝐺𝐡(𝑓(π‘₯0),𝑓𝑛(π‘₯),𝑓𝑛(π‘₯))∨𝐺𝐡(𝑓𝑛(π‘₯),π‘₯0,π‘₯0)=𝐺𝐡(𝑓(π‘₯0),𝑓(π‘“π‘›βˆ’1(x)),𝑓(π‘“π‘›βˆ’1𝐺(π‘₯)))∨𝐡(𝑓𝑛(π‘₯),π‘₯0,π‘₯0) as π‘›β†’βˆžwe deduce, 𝐺𝐡(𝑓(π‘₯0),𝑓(π‘“π‘›βˆ’1(π‘₯)),𝑓(π‘“π‘›βˆ’1(π‘₯)))∨𝐺𝐡(𝑓𝑛(π‘₯),π‘₯0,π‘₯0)β†’0, so the above equation will be 𝐺𝐡(𝑓(π‘₯0),π‘₯0,π‘₯0)≀0, therefore, 𝑓(π‘₯0)=π‘₯0.

Theorem 2.17. Let (𝑋,𝐺𝐡) be a 𝐺𝐡-metric space and π‘“βˆΆπ‘‹β†’π‘‹ be 𝐺𝐡-continuous. If for π‘₯βˆˆπ‘‹, the sequence {𝑓𝑛(π‘₯)} has a convergent subsequence {𝑓𝑛𝑖(π‘₯)}𝐺𝐡-converges to 𝑝, and 𝐺𝐡(𝑓𝑛𝑖(π‘₯),𝑓𝑛𝑖+1(π‘₯),𝑓𝑛𝑖+1(π‘₯))β†’0, then 𝑓 has a fixpoint.

Proof. Since {𝑓𝑛𝑖(π‘₯)}→𝑝, then by 𝐺𝐡-continuity of 𝑓 we have 𝑓(𝑓𝑛𝑖(π‘₯))=𝑓𝑛𝑖+1(π‘₯)→𝑓(𝑝), and we have 𝐺𝐡(𝑝,𝑓(𝑝),𝑓(𝑝))≀𝐺𝐡(𝑝,𝑓𝑛𝑖(π‘₯),𝑓𝑛𝑖(π‘₯))βˆ¨πΊπ΅ξ€·π‘“π‘›π‘–(π‘₯),𝑓𝑛𝑖+1(π‘₯),𝑓𝑛𝑖+1ξ€Έ(π‘₯)βˆ¨πΊπ΅ξ€·π‘“π‘›π‘–+1ξ€Έ.(π‘₯),𝑓(𝑝),𝑓(𝑝)(2.5) As π‘–β†’βˆž we get 𝐺𝐡(𝑝,𝑓(𝑝),𝑓(𝑝))≀0, which implies that 𝑓(𝑝)=𝑝.

Theorem 2.18. A 𝐺𝐡-metric space 𝑋 is 𝐺𝐡-Hausdorff.

Proof. Consider two elements π‘₯, 𝑦 belong to 𝑋 that π‘₯≠𝑦, and two subsets 𝐴={π‘₯βˆˆπ‘‹βˆ£πΊπ΅(π‘₯,π‘₯,𝑧)<𝐺𝐡(π‘₯,π‘₯,𝑦)} and 𝐡={π‘₯βˆˆπ‘‹βˆ£πΊπ΅(π‘₯,π‘₯,𝑦)<𝐺𝐡(π‘₯,π‘₯,𝑧)}. These are two disjoint neighbourhoods of π‘₯ and 𝑦, respectively. It is sufficient to prove that the subsets 𝐴 and 𝐡 are open. Suppose π‘‘βˆˆπ΄. Put 𝐺𝐡(𝑑,𝑑,𝑧)=π‘Ÿ1, 𝐺𝐡(𝑑,𝑑,𝑦)=π‘Ÿ2 and 𝛿=π‘Ÿ2βˆ’π‘Ÿ1. If π‘ βˆˆπ΅πΊπ΅(𝑑,𝛿), then 𝐺𝐡(𝑠,𝑠,𝑧)<𝐺𝐡(𝑑,𝑑,𝑧)∨𝐺𝐡(𝑠,𝑠,𝑑)<π‘Ÿ2=𝐺𝐡(𝑑,𝑑,𝑦)<𝐺𝐡(𝑑,𝑑,𝑠)∨𝐺𝐡(𝑠,𝑠,𝑦)=𝐺𝐡(𝑠,𝑠,𝑦), therefore, π‘ βˆˆπ΄ and 𝐡𝐺𝐡(𝑑,𝛿)βŠ‚π΄. This proves 𝑋 is Hausdorff.

Theorem 2.19. A 𝐺𝐡-metric space 𝑋 is 𝐺𝐡-normal.

Proof. Let 𝐺𝐡(π‘₯,π‘₯,Dβ‹€)={𝐺𝐡(π‘₯,π‘₯,𝑦)βˆ£π‘¦βˆˆπ·}. Consider two closed disjoint subsets 𝐸 and 𝐹 of 𝑋. Put 𝐴={π‘₯βˆˆπ‘‹βˆ£(π‘₯,π‘₯,𝐹)<𝐺𝐡(π‘₯,π‘₯,𝐸)} and 𝐡={π‘₯βˆˆπ‘‹πΊπ΅(π‘₯,π‘₯,𝐸)<𝐺𝐡(π‘₯,π‘₯,𝐹)}. It is clear that 𝐴,𝐡 are disjoint open subsets of 𝑋, πΈβŠ‚π΄, and πΉβŠ‚π΅.

Corollary 2.20. A 𝐺𝐡-metric space 𝑋 is 𝐺𝐡-regular.

2.2. Monoid Invariant 𝐺𝐡-Metric

Definition 2.21. A 𝐺𝐡-metric 𝑔𝑏 on a (𝑋,βŠ™) is π‘š-invariant (or monoid invariant) when 𝐺𝐡(π‘₯βŠ™π‘Ž,π‘¦βŠ™π‘Ž,π‘§βŠ™π‘Ž)≀𝐺𝐡(π‘₯,𝑦,𝑧) and 𝐺𝐡(π‘ŽβŠ™π‘₯,π‘ŽβŠ™π‘¦,π‘ŽβŠ™π‘§)≀𝐺𝐡(π‘₯,𝑦,𝑧) are valid for any π‘₯,𝑦,𝑧,π‘Žβˆˆπ‘‹.

Proposition 2.22. Let (𝑋,βŠ™) be a monoid. If 𝐺𝐡 is an m-invariant 𝐺𝐡-metric on X, then (1)𝐺𝐡(π‘₯βŠ™π‘Ž,π‘¦βŠ™π‘,π‘§βŠ™π‘)≀𝐺𝐡(π‘₯,𝑦,𝑧)∨𝐺𝐡(π‘Ž,𝑏,𝑐) and(2)𝐺𝐡(π‘₯1βŠ™π‘₯2βŠ™π‘₯3βŠ™β‹―βŠ™π‘₯𝑛,𝑦1βŠ™π‘¦2βŠ™π‘¦3βŠ™β‹―βŠ™π‘¦π‘›,𝑧1βŠ™π‘§2βŠ™π‘§3βŠ™β‹―βŠ™π‘§π‘›)≀𝐺𝐡(π‘₯1,𝑦1,𝑧1)∨𝐺𝐡(π‘₯2,𝑦2,𝑧2)∨𝐺𝐡(π‘₯3,𝑦3,𝑧3)βˆ¨β‹―βˆ¨πΊπ΅(π‘₯𝑛,𝑦𝑛,𝑧𝑛).

Proof. (1) We have𝑔𝑏(π‘₯βŠ™π‘Ž,π‘¦βŠ™π‘,π‘§βŠ™π‘)≀𝑔𝑏(π‘₯βŠ™π‘Ž,π‘¦βŠ™π‘Ž,π‘¦βŠ™π‘Ž)βˆ¨π‘”π‘(π‘¦βŠ™π‘Ž,π‘¦βŠ™π‘,π‘§βŠ™π‘)≀𝑔𝑏(π‘₯,𝑦,𝑦)βˆ¨π‘”π‘(π‘¦βŠ™π‘Ž,π‘¦βŠ™π‘,π‘¦βŠ™π‘)βˆ¨π‘”π‘(π‘¦βŠ™π‘,π‘¦βŠ™π‘,π‘§βŠ™π‘)≀𝑔𝑏(π‘₯,𝑦,𝑦)βˆ¨π‘”π‘(π‘Ž,𝑏,𝑐)βˆ¨π‘”π‘(𝑦,𝑦,𝑧)≀𝑔𝑏(π‘₯,𝑦,𝑧)βˆ¨π‘”π‘(π‘₯,𝑦,𝑧)βˆ¨π‘”π‘(π‘Ž,𝑏,𝑐)=𝑔𝑏(π‘₯,𝑦,𝑧)βˆ¨π‘”π‘(π‘Ž,𝑏,c).(2.6)
(2) It concludes by induction.

Put (𝑋,βŠ™) is a monoid and 𝐺𝐡 be a 𝐺𝐡-metric on 𝑋. We call 𝐺𝐡 is left effective on 𝑋 if 𝐺𝐡(π‘ŽβŠ™π‘₯,π‘ŽβŠ™π‘¦,π‘ŽβŠ™π‘§)=𝐺𝐡(π‘₯,𝑦,𝑧) for all π‘₯,𝑦,π‘§βˆˆπ‘‹ then π‘Ž=𝑒. And we say 𝐺𝐡 is left free 𝐺𝐡-metric on 𝑋 if for a triple (π‘₯,𝑦,𝑧)βˆˆπ‘‹Γ—π‘‹Γ—π‘‹, 𝐺𝐡(π‘ŽβŠ™π‘₯,π‘ŽβŠ™π‘¦,π‘ŽβŠ™π‘§)=𝐺𝐡(π‘₯,𝑦,𝑧) then π‘Ž=𝑒. The right effective (free) is defined samely, and we call 𝐺𝐡 is effective (free) if it is left and right effective (free) on 𝑋. Immediately we deduce, if 𝐺𝐡 is free 𝐺𝐡-metric on (𝑋,βŠ™) then it is effective 𝐺𝐡-metric.

Proposition 2.23. Put π‘“βˆΆ(𝑋,βŠ™)β†’(π‘‹ξ…ž,βŠ—) is a monoid homomorphism and πΊπ΅ξ…ž is a 𝐺𝐡-metric on π‘‹ξ…ž the following results are valid: (1)πΊπ΅βˆΆπ‘‹Γ—π‘‹Γ—π‘‹β†’π΅, (π‘₯,𝑦,𝑧)β†¦πΊξ…žπ΅(𝑓(π‘₯),𝑓(𝑦),𝑓(𝑧)) is a 𝐺𝐡-metric on (𝑋,βŠ™).(2)If πΊξ…žπ΅ is m-invariant, then 𝐺𝐡 is m-invariant.(3)The 𝐺𝐡-metric 𝐺𝐡 on 𝑋 is free (or effective) if πΊξ…žπ΅ is free (or effective) on π‘‹ξ…ž.

Let 𝐸 and 𝐹 be subsets of 𝑋. The set 𝐸 is congruent to 𝐹, written πΈβ‰ˆπΉ, provided there exist a map π‘“βˆΆπΈβ†’πΉ that for any π‘₯,𝑦,π‘§βˆˆπΈ, 𝐺𝐡(π‘₯,𝑦,𝑧)=𝐺𝐡(𝑓(π‘₯),𝑓(𝑦),𝑓(𝑧)) and that 𝐹 is congruent to 𝐸 by the map π‘“βˆ’1βˆΆπΉβ†’πΈ. The map 𝑓(π‘œπ‘Ÿπ‘“βˆ’1) is called a congruence between 𝐸 and 𝐹. Clearly every single set {π‘₯}, π‘₯βˆˆπ‘‹ and each map 𝑓, {π‘₯} congruent to {𝑓(π‘₯)}.

Definition 2.24. A motion of 𝑋 is a congruence of 𝑋 with itself.

Proposition 2.25. Let (𝑋,𝐺𝐡) be a 𝐺𝐡-metric space. The set β„³(𝑋)={π‘“βˆΆπ‘‹β†’π‘‹βˆ£π‘“ismotion} is a group.

Proof. The identity map belongs to β„³(𝑋) so β„³(𝑋) is not empty. Group action is composition of maps, so the identity map is the identity element. If π‘”βˆˆβ„³(𝑋), then 𝐺𝐡(π‘”βˆ’1(π‘₯),π‘”βˆ’1(𝑦),π‘”βˆ’1(𝑧))=𝐺𝐡(𝑔(π‘”βˆ’1(π‘₯)),𝑔(π‘”βˆ’1(𝑦)),𝑔(π‘”βˆ’1(𝑧)))=(π‘₯,𝑦,𝑧), so π‘”βˆ’1βˆˆβ„³(π‘₯).

Theorem 2.26. Let (𝑋,𝐺𝐡) be a 𝐺𝐡-metric space and π‘“βˆΆπ‘‹β†’π‘‹ is a motion of 𝑋, with the properties: if {𝑓𝑛}β†’π‘₯, then 𝐺𝐡(𝑓𝑛𝑖+1(π‘₯),𝑓𝑛𝑖+1(π‘₯),𝑓𝑛(π‘₯))β†’0, then the set of fixpoints of 𝑓 (or 𝐹(𝑓)) is equal to the set of fixpoints of 𝑓𝑛(or𝐹(𝑓𝑛)) for all π‘›βˆˆβ„•.

Proof. Clearly 𝐹(𝑓)βŠ†πΉ(𝑓𝑛). Suppose π‘₯∈𝐹(𝑓𝑛), so the sequence {π‘“π‘›π‘˜(π‘₯)}βˆžπ‘˜=1β†’π‘₯, because π‘“π‘˜π‘›(π‘₯)=π‘₯, for all π‘˜βˆˆβ„•. Now by Theorem 2.16, we conclude 𝑓(π‘₯)=π‘₯.

Proposition 2.27. Let (𝑋,βˆ—) be a monoid and 𝐺𝐡 be a 𝐺𝐡-metric on 𝑋. (1)If for all π‘Žβˆˆπ‘‹, πΏπ‘Ž and π‘…π‘Ž be motions of 𝑋, then 𝐺𝐡 is m-invariant.(2)πΏπ‘Ž (or π‘…π‘Ž) (π‘Žβ‰ π‘’) cannot be a motion of 𝑋, if 𝐺𝐡 is effective.

Example 2.28. The 𝐺𝐡-metric defined in Example 2.4 is invariant on the monoid (π•Ž,+), but it is not effective. Clearly for all π‘˜βˆˆβ„•, π‘“π‘˜βˆΆπ•Žβ†’π•Ž, 𝑛↦𝑛+π‘˜ is a motion.

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