aa r X i v : . [ m a t h . G M ] J un FUZZY GYRONORMS ON GYROGROUPS
LI-HONG XIE
Abstract.
The concept of gyrogroups is a generalization of groups which do not ex-plicitly have associativity. In this paper, the notion of fuzzy gyronorms on gyrogroupsis introduced. The relations of fuzzy metrics (in the sense of George and Veeramani),fuzzy gyronorms and gyronorms on gyrogroups are studied. Also, the fuzzy metricstructures on fuzzy normed gyrogroups are discussed. In the last, the fuzzy metriccompletion of a gyrogroup with an invariant metric are studied. We mainly show thatlet d be an invariant metric on a gyrogroup G and ( b G, b d ) is the metric completion ofthe metric space ( G, d ); then for any continuous t -norm ∗ , the standard fuzzy met-ric space ( b G, M b d , ∗ ) of ( b G, b d ) is the (up to isometry) unique fuzzy metric completionof the standard fuzzy metric space ( G, M d , ∗ ) of ( G, d ); furthermore, ( b G, M b d , ∗ ) is afuzzy metric gyrogroup containing ( G, M d , ∗ ) as a dense fuzzy metric subgyrogroupand M b d is invariant on b G . Applying this result, we obtain that every gyrogroup G with an invariant metric d admits an (up to isometric) unique complete metric space( b G, b d ) of ( G, d ) such that b G with the topology introduced by b d is a topology gyrogroupcontaining G as a dense subgyrogroup and b d is invariant on b G . Introduction
Taking as a point of starting the notion of a Menger space, Kramosil and Michalekintroduced a notion of metric fuzziness [18] which became an interesting and fruitfularea of research(see for example[14, 20, 21, 22]). Furthermore, fuzzy metric spaceshave been investigated by several authors from different points of view (see for example[8, 9, 16]). In particular, George and Veeramani [12], by modifying a definition of fuzzymetric space given by Kramosil and Michalek [18], have introduced and studied a newand interesting notion of a fuzzy metric space with the help of continuous t -norms.Recall that a binary operation ∗ : [0 , × [0 , → [0 ,
1] is a continuous t -norm [26] if ∗ satisfies the following conditions:(i) ∗ is associative and commutative;(ii) ∗ is continuous;(iii) a ∗ a for all a ∈ [0 , a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d , with a, b, c, d ∈ [0 , t -norms are ∧ , · and ∗ L (the Lukasiewicz t -norm), which are defined by a ∧ b = min { a, b } , a · b = ab and a ∗ L b = max { a + b − , } ,respectively. One can easily show that ∗ ≤ ∧ for every continuous t -norm ∗ . Definition 1.1. (in the sense of George and Veeramani [12]) A fuzzy metric on a set X is a pair ( M, ∗ ) such that M is a fuzzy set in X × X × (0 , + ∞ ) and ∗ is a continuous t -norm satisfying for all x, y, z ∈ X and t, s > M ( x, y, t ) > Key words and phrases.
Gyrogroups; Fuzzy gyronorms; Fuzzy metrics; Completions of fuzzy metrics;Fuzzy normed gyrogroup.This work is supported by Natural Science Foundation of China (Grant No.11871379, 11526158) andthe Innovation Project of Department of Education of Guangdong Province, China. (ii) M ( x, y, t ) = 1 if and only if x = y ;(iii) M ( x, y, t ) = M ( y, x, t );(iv) M ( x, y, t + s ) ≥ M ( x, z, t ) ∗ M ( z, y, s );(v) M ( x, y, − ) : (0 , + ∞ ) → [0 ,
1] is continuous.By a fuzzy metric space we mean an ordered triple (
X, M, ∗ ) such that X is a set and( M, ∗ ) is a fuzzy metric on X . It is known that every fuzzy metric ( M, ∗ ) on a set X induces a topology τ M on X , which has as a base the family of open sets of the form { B M ( x, ε, t ) : x ∈ X, ε ∈ (0 , , t > } , where B M ( x, ε, t ) = { y ∈ X : M ( x, y, t ) > − ε } for all x ∈ X , ε ∈ (0 , t > G, M, ∗ ) such that ( M, ∗ )is invariant on G is a fuzzy metric group (in the sense of Kramosil and Michalek)[24,Theorem 2.2].In [31], Ungar studies a parametrization of the Lorentz transformation group. Thisleads to the formation of gyrogroup theory, a rich subject in mathematics (among others,the interested reader can consult [1, 11]). Loosely speaking, a gyrogroup (see Definition2.1) is a group-like structure in which the associative law fails to satisfy. Recently,topological gyrogroups are studied by Atiponrat [3] and Cai et al [6] and so on. Inparticular, Cai et al [6] extended the famous Birkhoff-Kakutani theorem by proving thatevery first-countable Hausdorff topological gyrogroup is metrizable [6, Theorem 2.3].Recently, Suksumran [30] studied the normed gyrogroup. In particular, Suksumranproved that the normed gyrogroups are homogeneous and form left invariant metricspaces and derive a version of the Mazur-Ulam theorem. Also, Suksumran given certainsufficient conditions, involving the right-gyrotranslation inequality and Klee’s condition,for a normed gyrogroup to be a topological gyrogroup (see [30]).Those lead to the notion of a fuzzy normed gyrogroup is introduced in this paper. Wemainly study the fuzzy metrics structures and the fuzzy metrics completion on fuzzynormed gyrogroups. The paper is organized as follows. In Section 2, some basic facts anddefinitions are stated. Section 3 is devoted to study the fuzzy normed gyrogroups. Therelations of fuzzy gyronorms, fuzzy metrics and gyronorms on gyrogroups are studied.We mainly show that: Every fuzzy normed gyrogroup G has an invariant fuzzy metricunder left gyrotranslations on G and every gyrogroup G with an invariant fuzzy metricunder left gyrotranslations is a fuzzy normed gyrogroup (see Theorems 3.4, 3.5 and3.7). Also, some sufficient conditions, which make a fuzzy normed gyrogroup to bea topological gyrogroup, are found (see Theorem 3.11). In Section 4 we consider thefuzzy metric completion of an invariant metric gyrogroup by proving that let d be aninvariant metric on a gyrogroup G and ( b G, b d ) is the metric completion of the metric space( G, d ); then for any continuous t -norm ∗ , the standard fuzzy metric space ( b G, M b d , ∗ )of ( b G, b d ) is the (up to isometry) unique fuzzy metric completion of the standard fuzzymetric space ( G, M d , ∗ ) of ( G, d ); furthermore, ( b G, M b d , ∗ ) is a fuzzy metric gyrogroupcontaining ( G, M d , ∗ ) as a dense fuzzy metric subgyrogroup and M b d is invariant on b G UZZY GYRONORMS ON GYROGROUPS 3 (see Theorem 4.4). Applying this result, we obtain that every gyrogroup G with aninvariant metric d admits an (up to isometric) unique complete metric space ( b G, b d ) of( G, d ) such that b G with the topology introduced by b d is a topology gyrogroup containing G as a dense subgyrogroup and b d is invariant on b G (see Corollary 4.5).2. Basic facts and definitions
The concept of gyrogroups as a generalization of groups, is originated from the studyof c -ball of relativistically admissible velocities with Einstein velocity addition as men-tioned by Ungar in [31].Let G be a nonempty set, and let ⊕ : G × G → G be a binary operation on G . Thenthe pair ( G, ⊕ ) is called a groupoid. A function f from a groupoid ( G , ⊕ ) to a groupoid( G , ⊕ ) is said to be a groupoid homomorphism if f ( x ⊕ x ) = f ( x ) ⊕ f ( x ) for anyelements x , x ∈ G . In addition, a bijective groupoid homomorphism from a groupoid( G, ⊕ ) to itself will be called a groupoid automorphism. We will write Aut ( G, ⊕ ) forthe set of all automorphisms of a groupoid ( G, ⊕ ). Definition 2.1. [31, Definition 2.7] Let ( G, ⊕ ) be a nonempty groupoid. We say that( G, ⊕ ) or just G (when it is clear from the context) is a gyrogroup if the followings hold:( G
1) There is an identity element e ∈ G such that e ⊕ x = x for all x ∈ G. ( G
2) For each x ∈ G , there exists an inverse element ⊖ x ∈ G such that ⊖ x ⊕ x = e. ( G
3) For any x, y ∈ G , there exists an gyroautomorphism gyr[ x, y ] ∈ Aut( G, ⊕ ) suchthat x ⊕ ( y ⊕ z ) = ( x ⊕ y ) ⊕ gyr[ x, y ]( z )for all z ∈ G .( G
4) For any x, y ∈ G , gyr[ x ⊕ y, y ] = gyr[ x, y ].One can easily show that any gyrogroup has a unique two-sided identity e , and anelement a of the gyrogroup has a unique two-sided inverse ⊖ a . It is clear that everygroup satisfies the gyrogroup axioms (the gyroautomorphisms are the identity map) andhence is a gyrogroup. Conversely, any gyrogroup with trivial gyroautomorphisms formsa group. From this point of view, gyrogroups naturally generalize groups.Proposition 2.2 summarizes some algebraic properties of gyrogroups, which will proveuseful in studying topological and geometric aspects of gyrogroups in Sections 3 and 4. Proposition 2.2. ([28, 29]) Let ( G, ⊕ ) be a gyrogroup and a, b, c ∈ G . Then(1) ⊖ ( ⊖ a ) = a Involution of inversion(2) ⊖ a ⊕ ( a ⊕ b ) = b Left cancellation law(3) gyr[ a, b ]( c ) = ⊖ ( a ⊕ b ) ⊕ ( a ⊕ ( b ⊕ c )) Gyrator identity(4) ⊖ ( a ⊕ b ) = gyr[ a, b ]( ⊖ b ⊖ a ) cf. ( ab ) − = b − a − (5) ( ⊖ a ⊕ b ) ⊕ gyr[ ⊖ a, b ]( ⊖ b ⊕ c ) = ⊖ a ⊕ c cf. ( a − b )( b − c ) = a − c (6) gyr[ a, b ] = gyr[ ⊖ b, ⊖ a ] Even property(7) gyr[ a, b ] = gyr − [ b, a ] , the inverse of gyr[ b, a ] Inversive symmetryAs far as we known, Atiponrat is the first scholar who extended the idea of topologicalgroups to topological gyrogroups as gyrogroups with a topology such that its binaryoperation is jointly continuous and the operation of taking the inverse is continuous. LI-HONG XIE
Definition 2.3. [3, Definition 1] A triple (
G, τ, ⊕ ) is called a topological gyrogroup ifand only if(1) ( G, τ ) is a topological space;(2) ( G, ⊕ ) is a gyrogroup; and(3) The binary operation ⊕ : G × G → G is continuous where G × G is endowed withthe product topology and the operation of taking the inverse ⊖ ( · ) : G → G , i.e. x → ⊖ x , is continuous.If a triple ( G, τ, ⊕ ) satisfies the first two conditions and its binary operation is con-tinuous, we call such triple a paratopological gyrogroup [4]. Sometimes we will just saythat G is a topological gyrogroup (paratopological gyrogroup) if the binary operationand the topology are clear from the context.Clearly, every topological group is a topological gyrogroup. Hence, tt is natural toask for the existence of a topological gyrogroup which is not a topological gyrogroup.In fact, Atiponrat show that the M¨obius gyrogroup and the Einstein gyrogroups withthe standard topology are such examples [3, Examples 2 and 3]. For the sake of com-pleteness, we give one of Examples as follows. Example . [3, Example 2] Let D be the complex open unit disk { z ∈ C : | z | < } .Consider D with the standard topology. Next, we define a M¨obius addition ⊕ M : D × D → D to be a function such that a ⊕ M b = a + b ab for all a, b ∈ D . Then D with the operator ⊕ M is not a group, which has no associativity. However, ithas been proved in section 3.4 of [31] that ( D , ⊕ M ) is a gyrogroup where the gyroauto-morphism define as follows: for any a, b, c ∈ D gyr[ a, b ]( c ) = 1 + ab ab c. This gyrogroup is one of the most important examples of gyrogroups. It is called the
M¨obius gyrogroup . Moreover, 0 is the identity, and for any a ∈ D , we get that − a ∈ D such that − a ⊕ M a = 0. Furthermore, D with the standard topology, the operator ⊕ M and the inverse operator are continuous, so ( D , ⊕ M ) is a topology gyrogroup, but not atopological group or or a paratopological group.3. Fuzzy normed gyrogroups
In this section we shall introduced the notions of fuzzy gyronorm on gyrogroups.Also, the fuzzy metric structure and geometric structures of fuzzy normed gyrogroupsare studied. Let us begin with the following definition.
Definition 3.1. [30, Definition 3.1] (
Gyronorms ). Let ( G, ⊕ ) be a gyrogroup. Afunction k · k : G → R is called a gyronorm on G if the following properties hold:(1) k x k≥ x ∈ G and k x k = 0 if and only if x = e ; (positivity)(2) k ⊖ x k = k x k for each x ∈ G ; (invariant under taking inverses)(3) k x ⊕ y k≤k x k + k y k ; for each x, y ∈ G (subadditivity)(4) k gyr[ a, b ]( x ) k = k x k for each x, a, b ∈ G . (invariant under gyrations)In 2018, Suksumran [30] introduced the notions of gyronorms on gyrogroups andsaid that any gyrogroup with a gyronorm is called a normed gyrogroup . Also, someintersting metric and geometric structures of normed gyrogroups are established in [30].This leads us to introduce the notions of Fuzzy Gyronorms on gyrogroup as follows. UZZY GYRONORMS ON GYROGROUPS 5
Definition 3.2. ( Fuzzy Gyronorms ) Given a gyrogroup ( G, ⊕ ) and a continuous t -norm ∗ , the pair ( N, ∗ ) is called a fuzzy gyronorms on G if N is a fuzzy set of G × (0 , + ∞ )satisfying the following conditions for all x, y ∈ G and all t, s ∈ (0 , + ∞ ):( N N ( x, t ) > N x = e if and only if N ( x, t ) = 1;( N N ( ⊖ x, t ) = N ( x, t );( N N ( x ⊕ y, t + s ) ≥ N ( x, t ) ∗ N ( y, s );( N N ( x, − ) : (0 , + ∞ ) → [0 ,
1] is continuous;( N N (gyr[ a, b ])( x ) , t ) = N ( x, t ) for all a, b ∈ G .Given a gyrogroup G , a continuous t -norm ∗ and a fuzzy set N of G × (0 , + ∞ ), theordered triple ( G, N, ∗ ) is called a fuzzy normed gyrogroup if ( N, ∗ ) is a fuzzy gyronormon G .The relations of normed gyrogroups and fuzzy normed gyrogroups are shown as fol-lows. Proposition 3.3.
Let ( G, k · k ) be a normed gyrogroup and define a fuzzy set N k·k of G × (0 , + ∞ ) by N k·k ( x, t ) = tt + k x k for each x ∈ G and t >
0, then the ordered triple(
G, N k·k , ∗ ) is a fuzzy normed gyrogroup, where ∗ is a contnuous t -norm; Proof.
Now let us check ( N k·k , ∗ ) satisfies the conditions in Definition 3.2. In fact, itis enough to show that N k·k satisfies Definition 3.2 ( N N k·k by Definition 3.1.In fact, take any x, y ∈ G and t, s >
0. Then N k·k ( x ⊕ y, t + s ) = s + ts + t + k x ⊕ y k ≥ s + ts + t + k x k + k y k , since k x ⊕ y k≤k x k + k y k by Definition 3.1 (3). Thus, to show that N k·k ( x ⊕ y, t + s ) ≥ N k·k ( x, t ) ∗ N k·k ( y, s ) = tt + k x k ∗ ss + k y k , it is enough to show that s + ts + t + k x k + k y k ≥ min { tt + k x k , ss + k y k } , since ∗ is a continuous t -norm.Without loss of generality, we assume that tt + k x k ≥ ss + k y k , which is equivalent to k x k t ≤ k y k s . LI-HONG XIE
Hence, we have that s + ts + t + k x k + k y k ≥ min { tt + k x k , ss + k y k }⇔ s + ts + t + k x k + k y k ≥ ss + k y k⇔ s + t + k x k + k y k s + t ≤ s + k y k s ⇔ k x k + k y k s + t ≤ k y k s ⇔ s k x k + s k y k≤ s k y k + t k y k⇔ s k x k≤ t k y k⇔ k x k t ≤ k y k s . Hence, by the assumption, we have proved that N k·k ( x ⊕ y, t + s ) ≥ N k·k ( x, t ) ∗ N k·k ( y, s )holds for each x, y ∈ G and s, t > (cid:3) Proposition 3.3 shows that every normed gyrogroup can introduce a fuzzy normedgyrogroup, for example, on the Einstein gyrogroups and the M¨obiusg yrogroups thereare fuzzy gyronorms, since they are normed gyrogroups (see [30, Theorems 3.3 and3.4]). Theorems 3.4 and 3.7 reveals the relations of fuzzy metrics and fuzzy gyronormson gyrogroups.
Theorem 3.4.
Let ( G, N, ∗ ) be a fuzzy normed gyrogroup. Define M N ( x, y, t ) = N ( ⊖ x ⊕ y, t ) for all x, y ∈ G and t > . Then ( M, ∗ ) is a fuzzy metric on G .Proof. Suppose that (
G, N, ∗ ) is a fuzzy normed gyrogroup. Let us verify that the pair( M N , ∗ ) satisfies (i)-(v) in Definition 1.1.(i): According to the definition of M N , M N ( x, y, t ) = N ( ⊖ x ⊕ y, t ), so M N ( x, y, t ) > N ( ⊖ x ⊕ y, t ) > x = y , then M N ( x, y, t ) = N ( ⊖ x ⊕ y, t ) = N ( e, t ) = 1 for all t > N M N ( x, y, t ) = 1 for all t >
0, then N ( ⊖ x ⊕ y, t ) = 1 for all t >
0. Thus, byDefinition 3.2( N
2) we obtain that ⊖ x ⊕ y = e , so x = y by by the left cancellation law.(iii): By Proposition 2.2 (4) and Definition 3.2 ( N
3) and ( N
6) we obtain that M N ( x, y, t ) = N ( ⊖ x ⊕ y, t )= N ( ⊖ ( ⊖ x ⊕ y ) , t )= N (gyr[ ⊖ x, y ]( ⊖ y ⊕ x ) , t )= N ( ⊖ y ⊕ x ) , t )= M N ( y, x, t ) . (iv): Take any x, y, z ∈ G . Then Proposition 2.2 (5) and Definition 3.2 ( N N UZZY GYRONORMS ON GYROGROUPS 7 M N ( x, z, s + t ) = N ( ⊖ x ⊕ z, s + t )= N (( ⊖ x ⊕ y ) ⊕ gyr[ ⊖ x, y ]( ⊖ y ⊕ z ) , s + t ) ≥ N ( ⊖ x ⊕ y, s ) ∗ N (gyr[ ⊖ x, y ]( ⊖ y ⊕ z ) , t )= N ( ⊖ x ⊕ y, s ) ∗ N ( ⊖ y ⊕ z, t )= M N ( x, y, s ) ∗ M ( y, z, t ) . (v): It is obvious by Definition 3.2 ( N (cid:3) The fuzzy metric induced by a fuzzy gyronorm in Theorem 3.4 is called a fuzzygyronorm metric on G .Let ( X, M, ∗ ) and ( Y, N, ⋆ ) be two fuzzy metric spaces. Recall that a mapping from X to Y is called an isometry if for each x, y ∈ X and each t > M ( x, y, t ) = N ( f ( x ) , f ( y ) , t ). Two fuzzy metric spaces ( X, M, ∗ ) and ( Y, N, ⋆ ) are called isomet-ric if there is an isometry from X onto Y (see [13, Definitions 1 and 2]). Theorem 3.5.
Let ( G, N, ∗ ) be a fuzzy normed gyrogroup. Then the fuzzy gyronormmetric ( M N , ∗ ) with respect to ( N, ∗ ) is invariant under left gyrotranslation: M N ( a ⊕ x, a ⊕ y, t ) = M N ( x, y, t ) for all x, y, a ∈ G and t ∈ (0 , + ∞ ) . Hence, every left gyrotranslation of G is an isometryof ( G, M N , ∗ ) .Proof. Let a, x, y ∈ G . Recall that the left gyrotranslation by a , denote by L a , is definedby L a ( x ) = a ⊕ x for all x ∈ G . Next, we prove that the fuzzy gyronorm metric ( M N , ∗ )is invariant under L a . In fact, M N ( L a ( x ) , L a ( y ) , t ) = M N ( a ⊕ x, a ⊕ y, t )= N ( ⊖ ( a ⊕ x ) ⊕ ( a ⊕ y ) , t )= N (gyr[ a, x ]( ⊖ x ⊖ a ) ⊕ ( a ⊕ y ) , t ) by Proposition 2.2 (4)= N (( ⊖ x ⊖ a ) ⊕ gyr[ x, a ]( a ⊕ y ) , t ) by Proposition 2 . N N ( ⊖ x ⊕ y, t ) by Proposition 2.2 (5)= M N ( x, y, t ) . (cid:3) Corollary 3.6. If ( G, N, ∗ ) is a fuzzy normed gyrogroup, then G with the topology intro-duced by the fuzzy gyronorm metric with respect to ( N, ∗ ) is a left topological gyrogroup(every left gyrotranslation of G is continuous). Hence, G is homogeneous.Proof. The fact that G with the topology introduced by the fuzzy gyronorm metricwith respect is a left topological gyrogroup directly follows from Theorem 3.5. Take any x, y ∈ G . Then it is obvious that L y ( L ⊖ x ( x )) = y , so by Theorem 3.5 L y ◦ L ⊖ x is asrequired. Hence G is homogeneous. (cid:3) Theorem 3.7.
Let G be a gyrogroup with a fuzzy metric ( M, ∗ ) . If ( M, ∗ ) is invariantunder left gyrotranslation, that is, M ( a ⊕ x, a ⊕ y, t ) = M ( x, y, t ) for all a, x, y ∈ G and t > and define N M ( x, t ) = M ( e, x, t ) then ( N M , ∗ ) is a fuzzygyronorm on G that generates the same fuzzy metric. LI-HONG XIE
Proof.
Clearly, N M is a fuzzy set of G × (0 , + ∞ ). Let us check ( N M , ∗ ) satisfies theconditions in Definition 3.2.( N N M ( x, t ) = M ( e, x, t ) > x ∈ G and t > N x = e , then N M ( e, t ) = M ( e, e, t ) = 1 holds for each t >
0. Conversely, if N M ( x, t ) = M ( e, x, t ) = 1 holds for each t >
0, then x = e .( N M, ∗ ) is invariant under left gyrotranslation, we have that the equalities N M ( ⊖ x, t ) = M ( e, ⊖ x, t ) = M ( x, e, t ) = M ( e, x, t ) = N M ( x, t )hold for each t > N M, ∗ ) under leftgyrotranslation, we have that N M ( x ⊕ y, t + s ) = M ( e, x ⊕ y, t + s )= M ( ⊖ x, y, t + s ) ≥ M ( ⊖ x, e, t ) ∗ M ( e, y, s )= M ( e, x, t ) ∗ M ( e, y, s )= N M ( x, t ) ∗ N M ( y, s )hold for each x, y ∈ G and t, s > N N M, ∗ ) under left gyrotranslation, we have that N M (gyr[ a, b ]( x ) , t ) = N M ( ⊖ ( a ⊕ b ) ⊕ ( a ⊕ ( b ⊕ x ) , t )= M ( e, ⊖ ( a ⊕ b ) ⊕ ( a ⊕ ( b ⊕ x ) , t )= M ( a ⊕ b, a ⊕ ( b ⊕ x ) , t )= M ( b, b ⊕ x ) , t )= M ( e, x ) , t )= N M ( x, t )hold for each a, b, x ∈ G and t, s > N M , ∗ ) is a fuzzy gyronorm on G .Now according to Theorem 3.5, since ( M, ∗ ) is invariant under left gyrotranslation,we have that N M ( ⊖ x ⊕ y, t ) = M ( e, ⊖ x ⊕ y, t ) = M ( x, y, t )hold for each x, y ∈ G and t >
0. This implies that ( N M , ∗ ) generates the same fuzzymetric ( M, ∗ ). (cid:3) Theorem 3.8.
Let ( G, N, ∗ ) be a fuzzy normed gyrogroup. If α ∈ Aut G and N ( α ( x ) , t ) = N ( x, t ) for all x ∈ G and t > , then α is an isometry of ( G, M N , ∗ ) , where ( M N , ∗ ) isthe fuzzy gyronorm metric with respect to ( N, ∗ ) .Proof. By assumption, we have that M N ( α ( x ) , α ( y ) , t ) = N ( ⊖ α ( x ) ⊕ α ( y ) , t )= N ( ⊖ α ( x ⊕ y ) , t )= N ( ⊖ x ⊕ y, t )= M N ( x, y, t ) . (cid:3) UZZY GYRONORMS ON GYROGROUPS 9
Corollary 3.9. If ( G, N, ∗ ) is a fuzzy normed gyrogroup, then the gyroautomorphismsof G are isometries of ( G, M N , ∗ ) , where ( M N , ∗ ) is the fuzzy gyronorm metric withrespect to ( N, ∗ ) . We have studied the fuzzy metric and geometric structures on fuzzy normed gy-rogroups. Next, we shall study the topological structures on fuzzy normed gyrogroups.Some sufficient conditions which make gyrogroups with some topologies become topo-logical gyrogroups are found.
Theorem 3.10.
Let ( G, N, ∗ ) be a fuzzy normed gyrogroup and ( M N , ∗ ) the fuzzy gy-ronorm metric with respect to ( N, ∗ ) . Consider the following conditions. ( I ) Right-gyrotranslation inequality: M N ( x ⊕ a, y ⊕ a, t ) ≥ M N ( x, y, t ) for all x, y, a ∈ G and t > ; ( I ) ′ Klee’s condition: M N ( x ⊕ y, a ⊕ b, t + s ) ≥ M N ( x, a, t ) ∗ M N ( y, b, s ) for all x, y, a, b ∈ G and t, s > ; ( II ) Commutative-like condition: N (( a ⊕ x ) ⊕ gyr [ a, x ]( y ⊖ a ) , t ) = N ( x ⊕ y, t ) for all x, y, a ∈ G and t > ; ( II ) ′ Right-gyrotranslation invariant: M N ( x ⊕ a, y ⊕ a, t ) = M N ( x, y, t ) for all x, y, a ∈ G and t > .Then ( II ) ⇔ ( II ) ′ ⇒ ( I ) ′ ⇔ ( I ) .Proof. The implication ( II ) ′ ⇒ ( I ) ′ is obvious. Thus it is enough to show that ( II ) ⇔ ( II ) ′ and ( I ) ′ ⇔ ( I ).( I ) ⇔ ( I ) ′ . Assume that the right-gyrotranslation inequality holds. Then we havethat M N ( x ⊕ y, a ⊕ b, t + s ) ≥ M N ( x ⊕ y, x ⊕ b, s ) ∗ M N ( x ⊕ b, a ⊕ b, t )= M N ( y, b, s ) ∗ M N ( x ⊕ b, a ⊕ b, t ) ≥ M N ( x, a, t ) ∗ M N ( y, b, s ) . Conversely, M N ( x ⊕ a, y ⊕ a, t ) ≥ M N ( x, y, t ) ∗ M ( a, a, M N ( x, y, t ) . ( II ) ⇔ ( II ) ′ . Assume commutative-like condition holds. Let x, a, y ∈ G and t, s > M N ( x ⊕ a, y ⊕ a, t ) = N ( ⊖ ( x ⊕ a ) ⊕ ( y ⊕ a ) , t )= N (gyr[ x, a ]( ⊖ a ⊖ x ) ⊕ ( y ⊕ a ) , t )= N (( ⊖ a ⊖ x ) ⊕ gyr[ a, x ]( y ⊕ a ) , t )= N (( ⊖ a ⊖ x ) ⊕ gyr[ ⊖ a, ⊖ x ]( y ⊕ a ) , t )= N ( ⊖ x ⊕ y, t )= M N ( x, y, t ) . Conversely, N ( x ⊕ y, t ) = M N ( ⊖ x, y, t )= M N ( ⊖ x ⊖ a, y ⊖ a, t )= N ( ⊖ ( ⊖ x ⊖ a ) ⊕ ( y ⊖ a ) , t )= N (gyr[ ⊖ x, ⊖ a ]( a ⊕ x ) ⊕ ( y ⊖ a ) , t )= N (( a ⊕ x ) ⊕ gyr[ ⊖ a, ⊖ x ]( y ⊖ a ) , t )= N (( a ⊕ x ) ⊕ gyr[ a, x ]( y ⊖ a ) , t ) . (cid:3) Theorem 3.11.
Let ( G, N, ∗ ) be a fuzzy normed gyrogroup. If one of the conditions ( I ) and ( I ) ′ in Theorem 3.10 holds, then G is a topological gyrogroup endowed with thetopology induced by the fuzzy gyronorm metric ( M N , ∗ ) with respect to ( N, ∗ ) .Proof. Firstly, we shall prove that the operator ⊕ : G × G → G is continuous. Take any x, y ∈ G and any open neighborhood V of x ⊕ y . Then there are ǫ ∈ (0 ,
1) and t > B M N ( x ⊕ y, ǫ, t ) = { z ∈ G : M N ( x ⊕ y, z, t ) > − ǫ } ⊆ V . Since the ∗ is acontinuous t -norm, there is ǫ ∈ (0 ,
1) such that(1 − ǫ , ∗ (1 − ǫ , ⊆ (1 − ǫ, . Then B M N ( x, ǫ , t ) and B M N ( y, ǫ , t ) are open neighborhoods of x and y , respectively.Take any a ∈ B M N ( x, ǫ , t ) and b ∈ B M N ( y, ǫ , t ). Then by the conditions in Theorem3.10 (II), M N ( x ⊕ y, a ⊕ b, t ) ≥ M N ( x, a, t ∗ M N ( y, b, t > − ǫ, since M N ( x, a, t ) > − ǫ and M N ( y, b, t ) > − ǫ . This implies that B M N ( x, ǫ , t ⊕ B M N ( y, ǫ , t ⊆ B M N ( x ⊕ y, ǫ, t ) ⊆ V. Hence we prove that the operator ⊕ is continuous.Secondly, we shall prove that the operator ⊖ : G → G is continuous. Take any x ∈ G and any open neighborhood V of ⊖ x . Then there are ǫ ∈ (0 ,
1) and t > B M N ( ⊖ x, ǫ, t ) = { z ∈ G : M N ( ⊖ x, z, t ) > − ǫ } ⊆ V . Clearly, B M N ( x, ǫ, t ) is anopen neighborhood of x .Take any y ∈ B M N ( x, ǫ, t ). Then M N ( ⊖ x, ⊖ y, t ) = M N ( e, ⊖ y ⊕ x, t )= M N ( y, x, t )= M N ( x, y, t ) > − ǫ Hence ⊖ y ∈ B M N ( ⊖ x, ǫ, t ), this implies that ⊖ B M N ( x, ǫ, t ) ⊆ B M N ( ⊖ x, ǫ, t ). Thuswe have proved that the operator ⊖ is continuous. (cid:3) Completions of invariant standard fuzzy metrics on gyrogroups
In this section, we shall study the fuzzy metric completion of an invariant metric d ona gyrogroup G by showing that the gyrogroup operator ⊕ can be extended to the fuzzymetric completion ( b G, c M d , ∗ ) of the standard fuzzy metric space ( G, M d , ∗ ) of the metricspace ( G, d ) such that ( b G, c M d , ∗ ) become a gyrogroup containing G as a subgyrogroup.Let us begin with some definitions and terms. UZZY GYRONORMS ON GYROGROUPS 11
A sequence ( x n ) n ∈ N in a fuzzy metric space ( X, M, ∗ ) is said to be a Cauchy sequence provided that for each ε ∈ (0 ,
1) and t >
0, there exists n ∈ N such that M ( x n , x m , t ) > − ε whenever n, m ≥ n . A fuzzy metric space ( X, M, ∗ ) where every Cauchy sequenceconverges is called complete .Let ( X, M, ∗ ) be a fuzzy metric space. Then a fuzzy metric completion of ( X, M, ∗ )is a complete fuzzy metric space ( b X, c M , ∗ ) such that ( X, M, ∗ ) is isometric to a densesubspace of b X . Unfortunately, there is a fuzzy metric that does not admit any fuzzymetric completion [13, Example 2].Let ( X, d ) be a metric space. Denote by a ∧ b = min { a, b } for all a, b ∈ [0 , M d the fuzzy set defined on X × X × (0 , + ∞ ) by M ( x, y, t ) = tt + d ( x, y ) . Then (
X, M d , ∧ ) is a fuzzy metric space (see [12, Example 2.9 and Remark 2.10]).Since for any continuous t -norm ∗ , one can easily show that ∗ ≤ ∧ , ( X, M d , ∗ ) is alsoa fuzzy metric space for any continuous t -norm ∗ . We call this fuzzy metric ( M d , ∗ )induced by a metric d the standard fuzzy metric with respect to the continuous t -norm ∗ . Gregori and Romaguera proved that for every metric space ( X, d ) the standard fuzzymetric space (
X, M d , · ) with respect to the usual multiplication · on [0 ,
1] admits an(up to isometry) unique fuzzy metric completion, which is exactly the standard fuzzymetric space of the completion of (
X, d ) with respect to the continuous t -norm · [13,Proposition 1]. In fact, borrowing their skills we have the more general result: Proposition 4.1.
Let (
X, d ) be a metric space and ∗ a continuous t -norm. Then thestandard fuzzy metric space ( X, M d , ∗ ) with respect to ∗ admits an (up to isometry)unique fuzzy metric completion, which is exactly the standard fuzzy metric space of thecompletion of ( X, d ) with respect to the ∗ . Proof.
For any continuous t -norm ⋆ , it is known that ( X, M d , ⋆ ) is a fuzzy metric spaceabove. Also we have the following facts:(1) the topologies τ M d and τ d introduced by ( M d , ⋆ ) and d on X , respectively, aresame;(2) if ( X, d ) is a complete metric space, then (
X, M d , ⋆ ) is a complete fuzzy metricspace.In fact, clearly, a sequence ( x n ) n ⊆ X converges to x in ( X, τ M d ) if and only if ( x n ) n converges to x in ( X, τ d ). Since the space ( X, τ d ) and ( X, τ M d ) are first-countable, wehave that τ M d = τ d . Clearly, every Cauchy sequence ( x n ) n in ( X, M d , ⋆ ) is also a Cauchysequence in ( X, d ), so by fact (1) one can obtain the fact (2).Let ( b X, M b d , ∗ ) be the standard fuzzy metric space of the completion ( b X, b d ) of ( X, d ).Then ( b X, M b d , ∗ ) is a complete fuzzy metric space by the fact (2) and it is the uniquefuzzy metric completion of ( X, M d , ∗ ) (up to isometry). Indeed, since there is an isom-etry f from ( X, d ) onto a dense subspace of ( b X, b d ) and by fact (1), f ( X ) is dense in( b X, M b d , ∗ ). Furthermore, clearly, by the equality b d ( f ( x ) , f ( y )) = d ( x, y ), we have that M b d ( f ( x ) , f ( y ) , t ) = M d ( x, y, t ) for all x, y ∈ X and t >
0. So ( b X, M b d , ∗ ) is a fuzzymetric completion of ( X, M d , ∗ ), and, by [13, Lemma 1], it is unique up to isometry. (cid:3) Definition 4.2.
Let G be a gyrogroup with a fuzzy metric ( M, ∗ ) (resp. metric d ).We say ( M, ∗ ) (resp. d ) is invariant on G if ( M, ∗ ) (resp. d ) is invariant under leftgyrotranslations and right gyrotranslations, that is M ( a ⊕ x, a ⊕ y, t ) = M ( x, y, t ) = M ( x ⊕ a, y ⊕ a, t ) for all x, y, a ∈ G and t > d ( a ⊕ x, a ⊕ y ) = d ( x, y ) = d ( x ⊕ a, y ⊕ a ) for all x, y, a ∈ G ).Let G be a gyrogroup with a fuzzy metric ( M, ∗ ). We call ( G, M, ∗ ) is a fuzzy metricgyrogroup if G with the topology introduced by ( M, ∗ ) is a topological gyrogroup. Theorem 4.3.
Let G be a gyrogroup with an invariant fuzzy metric ( M, ∗ ) . Then ( G, M, ∗ ) is a fuzzy metric gyrogroup.Proof. Since ( M, ∗ ) is invariant on G , according to Theorem 3.7, G is a fuzzy normedgyrogroup with the corresponding fuzzy metric ( M, ∗ ). Hence G with the topologyintroduced by ( M, ∗ ) is a topological gyrogroup by Theorem 3.11. (cid:3) Theorem 4.4.
Let d be an invariant metric on a gyrogroup G and ( b G, b d ) is the metriccompletion of the metric space ( G, d ) . Then for any continuous t -norm ∗ , the standardfuzzy metric space ( b G, M b d , ∗ ) of ( b G, b d ) is the (up to isometry) unique fuzzy metric com-pletion of the standard fuzzy metric space ( G, M d , ∗ ) of ( G, d ) . Furthermore, ( b G, M b d , ∗ ) is a fuzzy metric gyrogroup containing ( G, M d , ∗ ) as a dense fuzzy metric subgyrogroupand M b d is invariant on b G .Proof. From Proposition 4.1 it follows that ( b G, M b d , ∗ ) is the (up to isometry) uniquefuzzy metric completion of the standard fuzzy metric space ( G, M d , ∗ ). Now we shallprove that ( b G, M b d , ∗ ) is a fuzzy metric gyrogroup containing ( G, ⊕ ) as a dense subgyrogroup and M b d is invariant on b G .Since M d ( x, y, t ) = tt + d ( x,y ) for each x, y ∈ G and t > d is invariant on G , so is( M d , ∗ ) on G .Take any points a, b ∈ b G . Consider two sequences ( a n ) n , ( b n ) n ⊆ G such thatlim n →∞ a n = a and lim n →∞ b n = b in ( b G, M b d , ∗ ). Then we claim that the sequence ( a n ⊕ b n ) n is a Cauchy sequence in ( G, M d , ∗ ), hence in ( b G, M b d , ∗ ). For this, fix ǫ ∈ (0 ,
1) and t >
0. Since ∗ is a continuous t -norm, there is s > − s ) ∗ (1 − s ) > − ǫ .Observing that ( a n ) n and ( b n ) n are Cauchy sequences in ( G, M d , ∗ ), thus there is an n ∈ ω such that M d ( a i , a j , t ) > − s and M ( b i , b j , t ) > − s whenever i, j > n . Since M d is invariant on G , we have that M d ( a i ⊕ b i , a j ⊕ b j , t ) ≥ M d ( a i ⊕ b i , a j ⊕ b i , t ∗ M d ( a j ⊕ b i , a j ⊕ b j , t M d ( a i , a j , t ∗ M d ( b i , b j , t − s ) ∗ (1 − s ) > − ǫ whenever i, j > n . Hence we have prove that the sequence ( a n ⊕ b n ) n is a Cauchysequence in ( b G, M b d , ∗ ).Now we define a binary operation b ⊕ on b G as follows: given two elements a, b ∈ b G and two sequences ( a n ) n , ( b n ) n ⊆ G such that lim n →∞ a n = a and lim n →∞ b n = b , a b ⊕ b = x ,where x is the limit of ( a n ⊕ b n ) n in ( b G, M b d , ∗ ).Let us show that b ⊕ is well defined. Choose two sequences ( c n ) n , ( d n ) n ⊆ G such thatlim n →∞ c n = a and lim n →∞ d n = b in ( b G, M b d , ∗ ). Then we claim that lim n →∞ ( c n ⊕ d n ) = x in( b G, M b d , ∗ ). UZZY GYRONORMS ON GYROGROUPS 13
In fact, take any ǫ ∈ (0 ,
1) and t >
0. Choose s ∈ (0 ,
1) such that (1 − s ) ∗ (1 − s ) ∗ (1 − s ) > − ǫ , then we have that n ∈ ω such that M b d ( x, a n ⊕ b n , t > − s,M b d ( a n , c n , t > − s,M b d ( b n , d n , t > − s hold whenever n > n , since lim j →∞ ( a j ⊕ b j ) = x , lim j →∞ d j = b = lim j →∞ b j and lim j →∞ a j = a = lim j →∞ c j hold in ( b G, M b d , ∗ ). Since M d is invariant on G , we have that M b d ( x, c n ⊕ d n , t ) ≥ M b d ( x, a n ⊕ b n , t ∗ M b d ( a n ⊕ b n , c n ⊕ d n , t ≥ M b d ( x, a n ⊕ b n , t ∗ M b d ( a n ⊕ b n , c n ⊕ b n , t ∗ M b d ( c n ⊕ b n , c n ⊕ d n , t M b d ( x, a n ⊕ b n , t ∗ M b d ( a n , c n , t ∗ M b d ( b n , d n , t ≥ (1 − s ) ∗ (1 − s ) ∗ (1 − s ) > − ǫ hold whenever n > n . Hence we have proved that lim n →∞ ( c n ⊕ d n ) = x in ( b G, M b d , ∗ ).Thus, the binary operation b ⊕ is well defined.We shall prove that ( ⊖ a n ) n is a Cauchy sequence in ( b G, M b d , ∗ ) for any sequence( a n ) n ⊆ G such that lim n →∞ a n = a in ( b G, M b d , ∗ ). Take any ǫ ∈ (0 ,
1) and t >
0. Sincelim n →∞ a n = a , there is n ∈ ω such that M b d ( a i , a j , t ) > − ǫ holds whenever i, j > n .Note that M d is invariant in G , so we have that M b d ( ⊖ a i , ⊖ a j , t ) = M d ( ⊖ a i , ⊖ a j , t )= M d ( a i , a j , t )= M b d ( a i , a j , t ) > − ǫ holds whenever i, j > n .Now we can define an operation b ⊖ on b G as follows: given an element a ∈ b G and asequence ( a n ) n ⊆ G such that lim n →∞ a n = a in ( b G, M b d , ∗ ), b ⊖ a = x , where x is the limitof ( ⊖ a n ) n in ( b G, M b d , ∗ ).Let us show that b ⊖ is well defined. Choose any sequence ( c n ) n ⊆ G such thatlim n →∞ c n = a in ( b G, M b d , ∗ ). Then we claim that lim n →∞ ( ⊖ c n ) = x in ( b G, M b d , ∗ ). Take any ǫ ∈ (0 ,
1) and t >
0. Then one can find s ∈ (0 ,
1) such that (1 − s ) ∗ (1 − s ) > − ǫ .Since lim j →∞ ( c j ) = a = lim j →∞ ( a j )and lim j →∞ ( ⊖ a j ) = x, we can find n ∈ ω such that M d ( a n , c n , t ) = M b d ( a n , c n , t ) > − s and M b d ( x, ⊖ a n , t ) > − s holds whenever n > n . Since that M d is invariant in G , we have that M b d ( x, ⊖ c n , t ) ≥ M b d ( x, ⊖ a n , t ∗ M b d ( ⊖ a n , ⊖ c n , t M b d ( x, ⊖ a n , t ∗ M d ( ⊖ a n , ⊖ c n , t M b d ( x, ⊖ a n , t ∗ M d ( a n , c n , t > (1 − s ) ∗ (1 − s ) > − ǫ holds whenever n > n . Hence we have prove that b ⊖ is well defined.Next, let us show that ( b G, b ⊕ ) is a gyrogroup. That is, the operator b ⊕ satisfies theaxioms G G e be the identity in G and take any a, b, c in b G and three sequences ( a n ) n , ( b n ) n , ( c n ) n ⊆ G such that lim n →∞ a n = a , lim n →∞ b n = b and lim n →∞ c n = c in ( b G, M b d , ∗ ).( G e b ⊕ b = b holds for each b ∈ b G . In fact, since ( G, ⊕ ) is a gyrogroup, we havethat e b ⊕ b = lim n →∞ ( e n ⊕ b n )= lim n →∞ b n = b, where e n = e holds for each n ∈ ω . Similarly, one can easily show that b b ⊕ e = b holdsfor each b ∈ b G .( G a ∈ b G , clearly,( b ⊖ a ) b ⊕ a = lim n →∞ ( ⊖ a n ) b ⊕ lim n →∞ a n = lim n →∞ ( ⊖ a n ⊕ a n ) = e = lim n →∞ ( a n ⊕ ( ⊖ a n )) = a b ⊕ ( b ⊖ a ) . ( G c gyr[ a, b ]( c ) = b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ c )) for each a, b, c ∈ b G .Firstly, let us verify that the equality ( ♣ ): a b ⊕ ( b b ⊕ c ) = ( a b ⊕ b ) b ⊕ c gyr[ a, b ]( c ) holds. Infact,( a b ⊕ b ) b ⊕ c gyr[ a, b ]( c ) = ( a b ⊕ b ) b ⊕ ( b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ c )))= ( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ ( b ⊖ ( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ ( lim n →∞ a n b ⊕ ( lim n →∞ b n b ⊕ lim n →∞ c n )))= lim n →∞ (( a n ⊕ b n ) ⊕ ( ⊖ ( a n ⊕ b n ) ⊕ ( a n ⊕ ( b n ⊕ c n ))))= lim n →∞ (( a n ⊕ b n ) ⊕ gyr[ a n , b n ]( c n ))= lim n →∞ ( a n ⊕ ( b n ⊕ c n ))= lim n →∞ a n b ⊕ ( lim n →∞ b n b ⊕ lim n →∞ c n )= a b ⊕ ( b b ⊕ c ) . Next, we shall prove that c gyr[ a, b ] ∈ Aut ( b G, b ⊕ ). Clearly, the mapping c gyr[ a, b ] isfrom ( b G, b ⊕ ) to itself. UZZY GYRONORMS ON GYROGROUPS 15
Firstly, we shall prove that c gyr[ a, b ] is a groupoid homomorphism. In fact, take any d ∈ b G and ( d n ) n ⊆ G such that lim n →∞ d n = d . Then we have that c gyr[ a, b ]( c b ⊕ d ) = b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ ( c b ⊕ d )))= b ⊖ ( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ ( lim n →∞ a n b ⊕ ( lim n →∞ b n b ⊕ ( lim n →∞ c n b ⊕ lim n →∞ d n )))= lim n →∞ ( ⊖ ( a n ⊕ b n ) ⊕ ( a n ⊕ ( b n ⊕ ( c n ⊕ d n ))))= lim n →∞ (gry[ a n , b n ]( c n ⊕ d n ))= lim n →∞ (gry[ a n , b n ]( c n ) ⊕ gry[ a n , b n ]( d n ))= lim n →∞ (( ⊖ ( a n ⊕ b n ) ⊕ ( a n ⊕ ( b n ⊕ c n ))) ⊕ ( ⊖ ( a n ⊕ b n ) ⊕ ( a n ⊕ ( b n ⊕ d n ))))= ( b ⊖ ( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ ( lim n →∞ a n b ⊕ ( lim n →∞ b n b ⊕ lim n →∞ c n ))) b ⊕ ( b ⊖ ( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ ( lim n →∞ a n b ⊕ ( lim n →∞ b n b ⊕ lim n →∞ d n )))= ( b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ c ))) b ⊕ ( b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ d )))= c gyr[ a, b ]( c ) b ⊕ c gyr[ a, b ]( d ) . Secondly, we shall prove that the equalities c gyr[ b ⊖ a, a ]( b ) = b, (1) b ⊖ a ( b ⊕ a b ⊕ b ) = b, (2)hold for each a, b ∈ b G .By the definitions of c gyr[ a, b ], b ⊖ and b ⊕ above, we have that c gyr[ b ⊖ a, a ]( b ) = b ⊖ ( b ⊖ a b ⊕ a ) b ⊕ ( b ⊖ a b ⊕ ( a b ⊕ b ))= b ⊖ ( lim n →∞ ( ⊖ a n ) b ⊕ lim n →∞ a n ) b ⊕ ( lim n →∞ ( ⊖ a n ) b ⊕ ( lim n →∞ a n b ⊕ lim n →∞ b n ))= lim n →∞ ( ⊖ ( ⊖ a n ⊕ a n ) ⊕ ( ⊖ a n ⊕ ( a n ⊕ b n )))= lim n →∞ (gyr[ ⊖ a n , a n ]( b n ))= lim n →∞ b n = b. By the equality ( ♣ ), (1) and ( G G
2) above, we have that b ⊖ a ( b ⊕ a b ⊕ b ) = ( b ⊖ a b ⊕ a ) b ⊕ c gyr[ b ⊖ a, a ]( b )= ( b ⊖ a b ⊕ a ) b ⊕ b = e b ⊕ b = b. Now we can prove that c gyr[ a, b ] is bijective. If c gyr[ a, b ]( c ) = c gyr[ a, b ]( d ), that is, b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ c )) = b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ d )) , then by (2) above one can easily obtain that c = d . This implies that the mapping c gyr[ a, b ] is injective. Next we claim the mapping c gyr[ a, b ] is onto. Take any c ∈ b G . Then c gyr[ a, b ]( c gyr[ b, a ]( c ))= c gyr[ a, b ]( b ⊖ ( b b ⊕ a ) b ⊕ ( b b ⊕ ( a b ⊕ c )))= b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ ( b ⊖ ( b b ⊕ a ) b ⊕ ( b b ⊕ ( a b ⊕ c )))))= b ⊖ ( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ ( lim n →∞ a n b ⊕ ( lim n →∞ b n b ⊕ ( b ⊖ ( lim n →∞ b n b ⊕ lim n →∞ a n ) b ⊕ ( lim n →∞ b n b ⊕ ( lim n →∞ a n b ⊕ lim n →∞ c n )))))= lim n →∞ ( ⊖ ( a n ⊕ b n ) ⊕ ( a n ⊕ ( b n ⊕ ( ⊖ ( b n ⊕ a n ) ⊕ ( b n ⊕ ( a n ⊕ c n ))))))= lim n →∞ (gyr[ a n , b n ]( ⊖ ( b n ⊕ a n ) ⊕ ( b n ⊕ ( a n ⊕ c n ))) by Proposition 2.2 (3)= lim n →∞ (gyr[ a n , b n ](gyr[ b n , a n ]( c n ))) by Proposition 2.2 (3)= lim n →∞ c n by Proposition 2.2 (7)= c. Hence we have proved that the mapping c gyr[ a, b ] is onto, furthermore, is bijective.( G a and b . For any c ∈ b G , we have that c gyr[ a b ⊕ b, b ]( c ) = b ⊖ (( a b ⊕ b ) b ⊕ b ) b ⊕ (( a b ⊕ b ) b ⊕ ( b b ⊕ c ))= b ⊖ (( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ lim n →∞ b n ) b ⊕ (( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ ( lim n →∞ b n b ⊕ lim n →∞ c n ))= lim n →∞ ( ⊖ (( a n ⊕ b n ) ⊕ b n ) ⊕ (( a n ⊕ b n ) ⊕ ( b n ⊕ c n )))= lim n →∞ (gyr[ a n ⊕ b n , b n ]( c n )) by Proposition 2.2 (3)= lim n →∞ (gyr[ a n , b n ]( c n )) by Definition 2.1 ( G n →∞ ( ⊖ ( a n ⊕ b n ) ⊕ ( a n ⊕ ( b n ⊕ c n )))= b ⊖ ( lim n →∞ a n b ⊕ lim n →∞ b n ) b ⊕ ( lim n →∞ a n b ⊕ ( lim n →∞ b n b ⊕ lim n →∞ c n ))= b ⊖ ( a b ⊕ b ) b ⊕ ( a b ⊕ ( b b ⊕ c ))= c gyr[ a, b ]( c ) . Hence, we have proved that c gyr[ a b ⊕ b, b ] = c gyr[ a, b ] holds for each a, b ∈ b G .Thus we have proved that ( b G, b ⊕ ) is a gyrogroup. Next, we shall prove that ( M b d , ∗ )is invariant on b G . Take any a, b, c ∈ b G and t >
0. Choose three sequences ( a n ) n , ( b n ) n ,( c n ) n ⊆ G such that lim n →∞ a n = a , lim n →∞ b n = b and lim n →∞ c n = c in ( b G, M b d , ∗ ). Then we UZZY GYRONORMS ON GYROGROUPS 17 have that M b d ( a b ⊕ b, a b ⊕ c, t ) = tt + b d ( a b ⊕ b, a b ⊕ c )= tt + b d ( lim n →∞ a n b ⊕ lim n →∞ b n , lim n →∞ a n b ⊕ lim n →∞ c n )= tt + b d ( lim n →∞ ( a n ⊕ b n ) , lim n →∞ ( a n ⊕ c n ))= tt + lim n →∞ b d ( a n ⊕ b n , a n ⊕ c n ) the metric b d is continuous on b G = tt + lim n →∞ b d ( b n , c n ) the metric b d is invariant on G = tt + b d ( lim n →∞ b n , lim n →∞ c n ) the metric b d is continuous on b G = tt + b d ( b, c )= M b d ( b, c, t ) . Similarly, one can prove that M b d ( b b ⊕ a, c b ⊕ a, t ) = M b d ( b, c, t ) . So ( M b d , ∗ ) is invariant on b G .Then from Theorem 4.3 it follows that b G with the topology introduced by M b d is atopological gyrogroup. Furthermore, ( b G, M b d , ∗ ) is a fuzzy metric gyrogroup containing G as a dense subgyrogroup, since one can easily show that a b ⊕ b = a ⊕ b for each a, b ∈ G . (cid:3) From Theorem 4.4 it follows:
Corollary 4.5.
Every gyrogroup G with an invariant metric d admits an (isometric)unique complete metric space ( b G, b d ) of ( G, d ) such that b G with the topology introducedby b d is a topology gyrogroup containing G as a dense subgyrogroup and b d is invarianton b G . Theorem 4.6.
Let ( G, M, ∗ ) be a fuzzy metric gyrogroup such that ( M, ∗ ) is invariantunder the left (right) gyrotranslation. If ( G, M, ∗ ) is a complete fuzzy metric, then everycompatible invariant under the left (right) gyrotranslation fuzzy metric on G is complete.Proof. Let ( N, ∗ ) be a compatible invariant under the left (right) gyrotranslation fuzzymetric on G . Take a Cauchy sequence ( x n ) n in ( G, N, ∗ ). Then we shall prove that( x n ) n is a Cauchy sequence in ( G, M, ∗ ). In fact, take any ε ∈ (0 ,
1) and t >
0. Sincethe topologies on G introduced by ( N, ∗ ) and ( M, ∗ ),respectively, are same, there is ε ∈ (0 ,
1) and t > B N ( e, ε , t ) ⊆ B M ( e, ε, t ), where e is the identityin G and B N ( e, ε , t ) = { x ∈ G : N ( e, x, t ) > − ε } , similar to B M ( e, ε, t ). Since( x n ) n is a Cauchy sequence in ( G, N, ∗ ), for ε and t > j ∈ ω such that N ( x i , x k , t ) > − ε whenever i, k > j . This implies that N ( e, ⊖ x i ⊕ x k , t ) > − ε ( N ( e, x k ⊕ ( ⊖ x i ) , t ) > − ε ), since N is invariant under the left (right) gyrotranslation.Hence ⊖ x i ⊕ x k ∈ B N ( e, ε , t ) ⊆ B M ( e, ε, t ) ( x k ⊕ ( ⊖ x i ) ∈ B N ( e, ε , t ) ⊆ B M ( e, ε, t )) whenever i, k > j . Note that M is invariant under the left (right) gyrotranslation on G , so M ( x i , x k , t ) > − ε whenever i, k > j . Thus we have proved that ( x n ) n is aCauchy sequence in ( G, M, ∗ ), so from the completion of M on G it follows that thesequence ( x n ) n converges in ( G, M, ∗ ). Since the topologies on G introduced by ( N, ∗ )and ( M, ∗ ),respectively, are same, the sequence ( x n ) n converges in ( G, N, ∗ ). Thus N isa complete fuzzy metric on G . (cid:3) Applying Theorems 4.4 and 4.6 we have the following result:
Corollary 4.7. If ( G, M, ∗ ) is a fuzzy metric gyrogroup such that ( M, ∗ ) is invari-ant, then every invariant under the left gyrotranslation fuzzy metric on the completion ( b G, c M , ∗ ) of ( G, M, ∗ ) is complete. References [1] T. Abe, K. Watanabe, Finitely generated gyrovector subspaces and orthogonal gyrodecompositionin the M¨obius gyrovector space, J. Math. Anal. Appl. (2017), 77-90.[2] C. Alegre, S. Romaguera, The Hahn-Banach extension theorem for fuzzy normed spaces revisited,Abstr. Appl. Anal. (2014), 151472.[3] W. Atiponrat, Topological gyrogroups: generalization of topological groups, Topol. Appl. (2017), 73-82.[4] W. Atiponrat, R. Maungchang, Complete regularity of paratopological gyrogroups, Topol. Appl. (2020), 106951.[5] T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. (2003),687-705.[6] Z. Cai, S. Lin, W. He, A note on paratopological loops, Bull. Malays. Math. Sci. Soc., DOI:10.1007/s40840-018-0616-y.[7] S.C. Cheng, J.N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. CalcuttaMath. Soc. (1994), 429-436.[8] Z.-K. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl. (1982), 74-95.[9] M.A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. (1979), 205-230.[10] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. (1992), 239-248.[11] M. Ferreira, Harmonic analysis on the M¨obius gyrogroup, J. Fourier Anal. Appl. (2) (2015),281-317.[12] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. (1994),395-399.[13] V. Gregori, S. Romaguera, On completion of fuzzy metric spaces, Fuzzy Sets Syst. (2002),399-404.[14] V. Gregori, A. Sapena, On fixed-point theorems in fuzzy metric space, Fuzzy Sets Syst. (2)(2002), 245-252.[15] J. Guti´errez-Garc´ıa, S. Romaguera, M. Sanchis, Standard fuzzy uniform structures based on con-tinuous t -norms, Fuzzy Sets Syst. (2012), 75-89.[16] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst. (1984), 215-229.[17] J.-H. Kim, G.A. Anastassiou, Ch. Park, Additive ρ -functional inequalities in fuzzy normed spaces,J. Comput. Anal. Appl. (2016), 1115-1126.[18] I. Kramosil, J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika (1975), 326-334.[19] K.Y. Lee, Approximation properties in fuzzy normed spaces, Fuzzy Sets Syst. (2016), 115-130.[20] S. Macario, M. Sanchis, Gromov-Hausdorff convergence of non-Archimedean fuzzy metric spaces,Fuzzy Sets Syst. (2015), 62-85.[21] G. Rano, T. Bag, S.K. Samanta, Some results on fuzzy metric spaces, J. Fuzzy Math. (4) (2011),925-938.[22] L.A. Ricarte, S. Romaguera, A domain-theoretic approach to fuzzy metric spaces, Topol. Appl. (2014), 149-159.[23] S. Romaguera, M. Sanchis, On fuzzy metric groups, Fuzzy Sets Syst. (2001), 109-115.[24] I. S´anchez, M. Sanchis, Complete invariant fuzzy metrics on groups, Fuzzy Sets Syst. (2017),http://dx.doi.org/10.2016/j.fss.2016.12.019. UZZY GYRONORMS ON GYROGROUPS 19 [25] I. S´anchez, M. Sanchis, Fuzzy quasi-pseudometrics on algebraic structures, Fuzzy Sets Syst. (2018), 79-86.[26] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math. (1960), 314-334.[27] H. Sherwood, On the completion of probabilistic metric spaces, Z. Wahrscheinlichkeitstheor. Verw.Geb. (1966), 62-64.[28] T. Suksumran, Analytic hyperbolic geometry and Albert Einstein’s Special Theory of Relativity,World Scientific, Hackensack, NJ, 2008.[29] T. Suksumran, Essays in mathematics and its applications: In honor of Vladimir Arnold, Th. M.Rassias and P. M. Pardalos (eds.), ch. The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’sTheorem, and Isomorphism Theorems, Springer, Switzerland, (2016), 369-437.[30] T. Suksumran, On metric structures of normed gyrogroups, arXiv:1810.10491v2[math.MG]3 Nov2018.[31] A.A. Ungar, Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity,World Scientific, 2008. (L.H. Xie) School of Mathematics and Computational Science, Wuyi University, Jiang-men 529020, P.R. China E-mail address ::