- Email: [email protected]
Fuzzy quasi-pseudometrics on algebraic structures
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.1 (1-8)
Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
Fuzzy quasi-pseudometrics on algebraic structures Iván Sánchez ∗,1 , Manuel Sanchis 2 Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Spain Received 29 January 2017; received in revised form 29 April 2017; accepted 23 May 2017
Abstract In this work we state a number of theorems about fuzzy (quasi-)pseudometrizable algebraic structures. Our most useful results are: (1) a fuzzy semitopological group whose topology is induced by a left-invariant fuzzy (quasi-)pseudometric, then it is a fuzzy (paratopological) topological group, (2) if the topology on a semigroup S is induced by an invariant fuzzy quasi-pseudometric, then S is a fuzzy topological semigroup, and (3) the same conclusion is valid for a left-invariant fuzzy quasi-pseudometric on a monoid G such that the left translations are open and the right translations are continuous at the identity e of G. By means of the standard fuzzy (quasi-)pseudometric Md associated to a (quasi-)pseudometric d, our results apply in the case of semitopological groups, semigroups and monoids in order to obtain new results that allow us to generalize and to strengthen previous outcomes. © 2017 Elsevier B.V. All rights reserved.
Keywords: Fuzzy (quasi-)pseudometric; Left topological group; Right topological group; Semitopological group; Paratopological group; Topological group; Topological semigroup; Invariant fuzzy (quasi-)pseudometric
1. Introduction In this paper we shall focus our attention on topological algebraic structures equipped with a fuzzy quasipseudometric in the sense of Kramosil and Michalek. Combinations of a fuzzy metric structure and an algebraic structure deserve special attention in fuzzy Topological Algebra. The most frequently studied structures fall into the so-called fuzzy normed spaces (among others, the interested reader can consult [2,4,15,16]), although fuzzy metric topological groups are also worthy of consideration (see [11,19,20]). In [8] Gregori and Romaguera remove the symmetric condition in the definition of a fuzzy metric (in the sense of Kramosil and Michalek, see [14]) and introduce the notion of a fuzzy quasi-metric space. This allows us to consider nonsymmetric structures which fit in the realm of fuzzy nonsymmetric topology: fuzzy quasi-metric spaces and fuzzy quasi-normed spaces ([1,5,7,10]). In this context,
* Corresponding author.
E-mail addresses: [email protected] (I. Sánchez), [email protected] (M. Sanchis). 1 The author was supported by CONACYT of Mexico, grant number 259783. 2 The second author is supported by the Spanish Ministerio de Economía y Competitividad (Grant MTM2016-77143-P), Generalitat Valenciana
(Grant AICO/2016/030) and Universitat Jaume I (Grant P1-1B2014-35). http://dx.doi.org/10.1016/j.fss.2017.05.022 0165-0114/© 2017 Elsevier B.V. All rights reserved.
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.2 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
2
the classical results in semigroups and paratopological groups related to (left-)invariant metrics (see, for example, [17, Theorem 2.1], [18, Proposition 3.2]) encourage enough to merit further investigation in fuzzy nonsymmetric topology. In this paper we deal with one of the oldest problems in Topological Algebra: find sufficient conditions in order that a topological algebraic structure (in particular a nonsymmetric structure) become a stronger topological structure (in particular, a symmetric structure). Our goal is to obtain conditions on fuzzy quasi-metrics (respectively, fuzzy quasi-pseudometrics) semitopological groups and fuzzy paratopological groups (respectively, on fuzzy topological semigroups) which imply that they are either fuzzy paratopological groups or topological groups. To be precise, we show the following: (1) if (G, M, ∗) is a fuzzy semitopological group whose topology τM is induced by a left-invariant fuzzy (quasi-)pseudometric M, then (G, M, ∗) is a fuzzy (paratopological) topological group, (2) if (M, ∗) is an invariant fuzzy quasi-pseudometric on a semigroup S, then (S, M, ∗) is a fuzzy topological semigroup, and (3) the same conclusion is valid when (M, ∗) is a left-invariant fuzzy quasi-pseudometric on a monoid G such that the left translations are open and the right translations are continuous at the identity e of G. It is worth mentioning that, by means of the standard fuzzy (quasi-)pseudometric Md associated to a (quasi-)pseudometric d, our results apply in the case of semitopological groups, semigroups and monoids in order to obtain new results that allow us to generalize and to strengthen previous outcomes by Liu [17, Theorem 2.1] and Ravsky [18, Proposition 3.2]. A celebrated theorem by Birkhoff–Kakutani says that a Hausdorff topological group is metrizable if and only if it is first-countable. It is also a well-known result that a first-countable paratopological group is quasi-metrizable (see [18]). In spite of these results, a first-countable topological semigroup need not be quasi-pseudometrizable (see [12]). 2. Preliminaries According to [21], a continuous t-norm is a binary operation ∗ : [0, 1] × [0, 1] → [0, 1] which satisfies the following conditions: (i) ∗ is associative and commutative, (ii) ∗ is continuous, (iii) a ∗ 1 = a for every a ∈ [0, 1], and (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, with a, b, c, d ∈ [0, 1]. It is a well-known fact, and easy to check, that for each continuous t-norm ∗ one has ∗ ≤ ∧, where ∧ is the continuous t-norm given by a ∧ b = min{a, b}. The interested reader is referred to [13] for further information on (continuous) t-norms. Following [8], a fuzzy quasi-pseudometric (in the sense of Kramosil and Michalek) 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 > 0: (i) (ii) (iii) (iv)
M(x, y, 0) = 0; M(x, x, t) = 1; M(x, z, t + s) ≥ M(x, y, t) ∗ M(y, z, s); M(x, y, _) : [0, +∞) → [0, 1] is left continuous.
A fuzzy pseudometric on X is a fuzzy quasi-pseudometric (M, ∗) on X which satisfies: (v) M(x, y, t) = M(y, x, t) for all x, y ∈ X and t > 0. By a fuzzy (quasi-)pseudometric space (in the sense of Kramosil and Michalek) we mean a triple (X, M, ∗) such that X is a set and (M, ∗) is a fuzzy (quasi-)pseudometric on X. Every fuzzy (quasi)-pseudometric (M, ∗) on a set X induces a topology τM on X having as a base the family {BM (x, ε, t) : x ∈ X, ε ∈ (0, 1), t > 0}, where BM (x, ε, t) = {y ∈ X : M(x, y, t) > 1 − ε} for all x ∈ X, ε ∈ (0, 1) and t > 0. Let (X, d) be a (quasi-)pseudometric space. Define a fuzzy set Md in X × X × [0, ∞) by ⎧ t ⎨ for all x, y ∈ X and t > 0; t + d(x, y) Md (x, y, t) = ⎩ 0 for all x, y ∈ X and t = 0. Then (Md , ∧) is a fuzzy (quasi-)pseudometric on X, and hence (Md , ∗) is a fuzzy quasi-pseudometric on X for all continuous t-norm ∗, the so-called (fixed a t-norm ∗ on X), the standard fuzzy (quasi-)pseudometric induced by d on X. It is known that the topology τM and the topology τd , induced by the (quasi-)pseudometric d, coincide (see [6,8,9]).
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.3 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
3
We now move on to notions from (fuzzy) Topological Algebra. Let G be an algebraic semigroup. Pick x ∈ G. The function λx : G → G defined by λx (g) = xg is called the left translation of G by x. Similarly, ρx : G → G defined as ρx (g) = gx is known as the right translation of G by x. A topological semigroup (G, τ ) is an algebraic semigroup G with a topology τ that makes the multiplication in G jointly continuous. A paratopological group G is a topological semigroup such that G is an algebraic group. We say that a paratopological group (G, τ ) is a topological group if the inverse is continuous, that is, if g −1 stands for the inverse element of g ∈ G, then the function g → g −1 from G onto G is continuous. (G, τ ) is said to be a left (respectively, right) topological group if the translations λx (respectively, ρx ) are continuous in (G, τM ) for all x ∈ G. The following result is a well known internal characterization of a (para)topological group. Theorem 2.1. ([3, Theorem 1.3.12]) Let G be a group with identity e and U a family of subsets of G containing e. If U satisfies the following conditions: (i) (ii) (iii) (iv)
for every U, V ∈ U, there exists W ∈ U such that W ⊆ U ∩ V ; for every U ∈ U and x ∈ U , there is V ∈ U such that V x ⊆ U ; for every U ∈ U and x ∈ G, we can find V ∈ U satisfying xV x −1 ⊆ U ; for every U ∈ U, there exists V ∈ U such that V 2 ⊆ U ;
then the family {U x : x ∈ G, U ∈ U} is a base for a topology τU on G. With this topology, G is a paratopological group, and the family {xU : x ∈ G, U ∈ U} is a base for the same topology on G. In addition, if U satisfies (v) for every U ∈ U, we can find V ∈ U with V −1 ⊆ U ; then (G, τU ) is a topological group. We will introduce the fuzzy structures we deal with Definition 2.2. By a fuzzy (quasi)-pseudometric semigroup we mean a triple (G, M, ∗) such that (G, M, ∗) is a fuzzy (quasi)-pseudometric space and (G, τM ) is a topological semigroup. A fuzzy (quasi)-pseudometric paratopological group is a fuzzy (quasi)-pseudometric semigroup (G, M, ∗) such that G is an algebraic group. We now address the last fuzzy structure we need: Definition 2.3. By a fuzzy (quasi)-pseudometric right topological group we mean a triple (G, M, ∗) such that (G, M, ∗) is a fuzzy (quasi)-pseudometric space and (G, τM ) is a right topological group. Fuzzy (quasi)-pseudometric left topological groups are defined in a similar way. If no confusion can arise, we write G for (G, τM ). The following notion plays a significant role in our results. Definition 2.4. A fuzzy quasi-pseudometric (M, ∗) on a semigroup G is left-invariant (respectively, right-invariant) if M(x, y, t) = M(ax, ay, t) (respectively, M(x, y, t) = M(xa, ya, t)) whenever a, x, y ∈ G and t > 0. We say that (M, ∗) is invariant if it is both left-invariant and right-invariant. Our notation and terminology are standard. For notions on Topological Algebra not defined here, the interested reader can consult [3]. 3. The results Our first result is concerning with the continuity of the operation on right fuzzy quasi-pseudometric topological groups. It provides a sufficient condition to obtain a fuzzy paratopological group. Theorem 3.1. If (G, M, ∗) is a fuzzy quasi-pseudometric right topological group such that (M, ∗) is left-invariant, then (G, M, ∗) is a fuzzy paratopological group.
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.4 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
4
Proof. Let e be the identity of G. According to [19], U = {B(e, n1 , n1 ) : n ∈ N} is a local base at e. Let us show that U = {B(e, n1 , n1 ) : n ∈ N} satisfies conditions (i)–(iv) in Theorem 2.1, that is, the topology τU associated to the family U makes G into a paratopological group. Item (i) follows from the fact that U is a local base at e in (G, τM ). To prove (ii) take n ∈ N and x ∈ B(e, n1 , n1 ). Since ρx is continuous at e and ρx (e) = x ∈ B(e, n1 , n1 ), there exists m ∈ N such that 1 1 1 1 1 1 , )) = B(e, , )x ⊆ B(e, , ). m m m m n n Thus, (ii) holds. Now, before to show (iii), we will prove that, for each n ∈ N and x ∈ G, we have: ρx (B(e,
1 1 1 1 xB(e, , ) = B(x, , ). n n n n
(1)
Indeed, take y ∈ B(e, n1 , n1 ). Since (M, ∗) is left-invariant, we have 1 1 1 M(x, xy, ) = M(e, y, ) > 1 − n n n so that xB(e, n1 , n1 ) ⊆ B(x, n1 , n1 ). For the other inclusion, pick z ∈ B(x, n1 , n1 ). Again, since (M, ∗) is left-invariant, we conclude that 1 1 1 M(e, x −1 z, ) = M(x, z, ) > 1 − . n n n This proves that x −1 z ∈ B(e, n1 , n1 ). Thus, z ∈ xB(e, n1 , n1 ) which shows (1). Let us show (iii). Pick n ∈ N and x ∈ G. Note that every right translation is a homeomorphism. So B(e, n1 , n1 )x is an open neighborhood of x. Hence there is m ∈ N such that B(x, m1 , m1 ) ⊆ B(e, n1 , n1 )x. This and (1) imply 1 1 −1 1 1 1 1 , )x = B(x, , )x −1 ⊆ B(e, , ). m m m m n n This proves (iii). Finally, we show (iv). Choose n ∈ N. Since the t-norm ∗ is continuous, there is k ∈ N such that for each q ≥ k we have xB(e,
(1 −
1 1 1 ) ∗ (1 − ) > 1 − . q q n
Put m = max{k, 2n}. Then, for each y, z ∈ B(e, m1 , m1 ), the following inequalities hold 1 2 1 1 M(e, yz, ) ≥ M(e, yz, ) ≥ M(e, y, ) ∗ M(y, yz, ) = n m m m 1 1 1 1 1 M(e, y, ) ∗ M(e, z, ) > (1 − ) ∗ (1 − ) > 1 − . m m m m n Therefore, B(e, m1 , m1 )B(e, m1 , m1 ) ⊆ B(e, n1 , n1 ). This proves (iv). By Theorem 2.1, (G, τU ) is a paratopological group and {xB(e, n1 , n1 ) : x ∈ G, n ∈ N} is a base for τU . Notice that equation (1) implies that {xB(e, n1 , n1 ) : x ∈ G, n ∈ N} also is a base for τM so that τU = τM . This shows that (G, M, ∗) is a fuzzy paratopological group. 2 If we replace fuzzy quasi-pseudometric by fuzzy pseudometric, we obtain a symmetric structure. Theorem 3.2. If (G, M, ∗) is a fuzzy pseudometric right topological group such that (M, ∗) is left-invariant, then (G, M, ∗) is a fuzzy topological group. Proof. By Theorem 3.1, (G, M, ∗) is a fuzzy paratopological group. We only need to prove that the family U = {B(e, n1 , n1 ) : n ∈ N} satisfies (v) of Theorem 2.1. Choose n ∈ N. If we take x ∈ B(e, n1 , n1 ), then
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.5 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
5
1 1 1 1 M(e, x −1 , ) = M(x, e, ) = M(e, x, ) > 1 − . n n n n We conclude that x −1 ∈ B(e, n1 , n1 ). So B(e, n1 , n1 ) is symmetric for every n ∈ N. It follows that (G, M, ∗) is a fuzzy topological group. 2 Arguing as in Theorems 3.1 and 3.2 we can obtain. Theorem 3.3. If (G, M, ∗) is a fuzzy quasi-pseudometric left topological group such that (M, ∗) is right-invariant, then (G, M, ∗) is a fuzzy paratopological group. If in addition, (G, M, ∗) is a fuzzy pseudometric left topological group, then (G, M, ∗) is a topological group. As far we know, the following four corollaries are new results on the classical theory. Corollary 3.4. Suppose that (G, τ ) is a left (right) topological group whose topology τ is induced by a right-(left-)invariant quasi-pseudometric. Then (G, τ ) is a paratopological group. Proof. Suppose that the topology τ is induced by a right-(left-)invariant quasi-pseudometric. Then the fuzzy quasipseudometric (Md , ∧) induced by d is right-(left-)invariant. Theorems 3.1 and 3.3 imply that (G, τM ) is a paratopological group. Since τM = τd = τ , we have that (G, τ ) is a paratopological group. 2 Remark 3.5. A topological group is balanced if it has a local base at the identity whose elements are invariant open sets. It is easy to see that if a fuzzy metric group G admits an invariant fuzzy metric, then G is balanced. The general linear group GL(2, R) is a fuzzy metric group, but it is not balanced (see [3, Exercise 1.8.d]). Hence GL(2, R) does not admit an invariant fuzzy metric. An argument similar to the one used in Corollary 3.4 allows us to obtain the following three results. Recall that a semitopological group G is a group G with a topology that makes the multiplication separately continuous. Note that a semitopological group is both a left and right topological group. Corollary 3.6. Suppose that (G, τ ) is a semitopological group whose topology τ is induced by a right-(or left-)invariant quasi-pseudometric d. Then (G, τ ) is a paratopological group. According to [17, Theorem 2.1], G is a metrizable topological group provided that G is a quasi-metrizable paratopological group with respect to a left continuous, left-invariant quasi-metric d where left-continuous means that the function d(g, _) is continuous for all g ∈ G. Notice that by Corollary 3.6 a semitopological group enjoying these properties is a paratopological group. Thus, we can extend [17, Theorem 2.1] in the following way: Corollary 3.7. Suppose that (G, τ ) is a semitopological group whose topology τ is induced by a left continuous, left-invariant quasi-metric. Then (G, τ ) is a metrizable topological group. Ravsky proved that if a paratopological group G is metrizable by a left-invariant metric, then G is a topological group (see [18, Proposition 3.2]). We now present two generalizations of this fact as a direct consequence of Theorems 3.1, 3.2 and 3.3. Corollary 3.8. Suppose that G is a left (right) topological group whose topology is induced by a right-(left-)invariant pseudometric. Then G is a topological group. It is easy to see that the topology on the Sorgenfrey line S is induced by the invariant quasi-metric d defined as d(x, y) = y − x if x ≤ y and d(x, y) = 1 if x > y. It is known that S is a paratopological group, but S is not a topological group. This shows that in Corollary 3.8, we cannot replace pseudometric by quasi-metric. Corollary 3.9. Suppose that (G, τ ) is a semitopological group whose topology τ is induced by a right-(or left-)invariant pseudometric d. Then (G, τ ) is a topological group.
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.6 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
6
We now move on to fuzzy semigroups. It is worth mentioning that Heath provides in [12] a valuable collection of nonmetrizable, (so, of nonfuzzy metrizable) first countable cancellative topological semigroups (for example, a nonmetrizable locally compact, cancellative abelian topological semigroup that is a quasimetrizable Moore space [12, Example 3.1]). Theorem 3.10. Let (M, ∗) be a fuzzy quasi-pseudometric on a semigroup S. If (M, ∗) is invariant, then (S, M, ∗) is a fuzzy topological semigroup. Proof. Take y, z ∈ S. Choose n ∈ N. Let k ∈ N be such that for each q ≥ k, (1 − q1 ) ∗ (1 − q1 ) > 1 − n1 . Put m = max{k, 2n}. If a ∈ B(y, m1 , m1 ) and b ∈ B(z, m1 , m1 ), then we have 1 2 1 1 M(yz, ab, ) ≥ M(yz, ab, ) ≥ M(yz, yb, ) ∗ M(yb, ab, ) = n m m m 1 1 1 1 1 M(z, b, ) ∗ M(y, a, ) ≥ (1 − ) ∗ (1 − ) > 1 − . m m m m n Therefore B(y, m1 , m1 )B(z, m1 , m1 ) ⊆ B(yz, n1 , n1 ). We have thus proved that multiplication is continuous in (G, τM ), i.e., (G, M, ∗) is a fuzzy topological semigroup. 2 Let us recall that a monoid is a semigroup with a neutral element. Theorem 3.11. Suppose that (M, ∗) is a left-invariant fuzzy quasi-pseudometric on a monoid G such that for each x ∈ G, λx is open and ρx is continuous at the identity e of (G, M, ∗). Then (G, M, ∗) is a fuzzy topological semigroup. Proof. Let e be the identity of G. We claim that for each n ∈ N and x ∈ G we have 1 1 1 1 xB(e, , ) ⊆ B(x, , ). n n n n
(2)
Indeed, take y ∈ B(e, n1 , n1 ). Since (M, ∗) is left-invariant, we have 1 1 1 M(x, xy, ) = M(e, y, ) > 1 − . n n n This proves (2). As a consequence of (2), we have that left translations are continuous at e. Now, we will show that for every n ∈ N, there is m ∈ N satisfying B(e,
1 1 1 1 1 1 , )B(e, , ) ⊆ B(e, , ). m m m m n n
(3)
For this, notice that, since the t -norm is continuous, there is k ∈ N such that for each q ≥ k we have that (1 − q1 ) ∗ (1 − q1 ) > 1 − n1 . Put m = max{k, 2n}. Then, for each y, z ∈ B(e, m1 , m1 ), the following inequalities hold: 1 2 1 1 M(e, yz, ) ≥ M(e, yz, ) ≥ M(e, y, ) ∗ M(y, yz, ) = n m m m 1 1 1 1 1 M(e, y, ) ∗ M(e, z, ) > (1 − ) ∗ (1 − ) > 1 − . m m m m n Therefore B(e, m1 , m1 )B(e, m1 , m1 ) ⊆ B(e, n1 , n1 ). Now, we will prove that the multiplication is continuous in (G, τM ). Take x, y ∈ G and n ∈ N. By (2), we have xyB(e, n1 , n1 ) ⊆ B(xy, n1 , n1 ). It follows from (3) that B(e, m1 , m1 )B(e, m1 , m1 ) ⊆ B(e, n1 , n1 ) for some m ∈ N. Hence xyB(e,
1 1 1 1 1 1 1 1 , )B(e, , ) ⊆ xyB(e, , ) ⊆ B(xy, , ). m m m m n n n n
(4)
By hypothesis, left translations are open. So yB(e, m1 , m1 ) is an open set in (G, τM ) which contains y. Also by hypothesis, ρy is continuous at e. Hence there is k ∈ N satisfying
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.7 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
1 1 1 1 1 1 ρy (B(e, , )) = B(e, , )y ⊆ yB(e, , ). k k k k m m Therefore (4)–(5) imply
7
(5)
1 1 1 1 1 1 1 1 1 1 xB(e, , )yB(e, , ) ⊆ xyB(e, , )B(e, , ) ⊆ B(xy, , ). k k m m m m m m n n Since left translations are open, xB(e, 1k , 1k ) and yB(e, m1 , m1 ) are open neighborhoods of x and y, respectively. Therefore, multiplication in (G, τM ) is continuous. This completes the proof. 2 Applying the previous results, we get the following results in semigroups and topological monoids. Corollary 3.12. Suppose that d is a invariant quasi-pseudometric on a semigroup S. Then (S, d) is a topological semigroup. Question 3.13. Suppose that d is a left-invariant pseudometric on a semigroup S. Is (S, d) a topological semigroup? Corollary 3.14. Suppose that d is a left-invariant quasi-pseudometric on a monoid G such that for each x ∈ G, λx is open and ρx is continuous at the identity e of (G, d). Then (G, d) is a topological semigroup. 4. Conclusion We study how the existence of a (left-)invariant fuzzy (quasi-)pseudometric on an algebraic-topological structure (like a fuzzy left topological group or a semigroup) implies that actually we have a stronger structure (like a fuzzy (para)topological group). Our results fit in a long tradition of research in Topological Algebra. When applied to classical structures, they allow us to obtain new outcomes that generalizes helpful results of the theory. Moreover, since a fuzzy (quasi)uniformity is defined by a suitable family of fuzzy (quasi)pseudometrics, starting from the results presented in the paper, a future area of research is under what conditions on the natural fuzzy quasi-uniformities on a fuzzy topological semigroup (respectively, a fuzzy paratopological group) we can obtain a stronger fuzzy structure. Acknowledgements The authors wish to thank the referees for their valuable comments and suggestions for the improvement of this paper. References [1] C. Alegre, S. Romaguera, On the uniform boundedness theorem in fuzzy quasi-normed spaces, Fuzzy Sets Syst. 282 (2016) 143–153. [2] C. Alegre, S. Romaguera, The Hahn–Banach extension theorem for fuzzy normed spaces revisited, Abstr. Appl. Anal. (2014) 151472, 7 pp. [3] A.V. Arhangel’skii, M.G. Tkachenko, Topological Groups and Related Structures, Atlantis Ser. Math., vol. I, Atlantis Press and World Scientific, Paris–Amsterdam, 2008. [4] T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, Ann. Fuzzy Math. Inform. 6 (2) (2013) 271–283. [5] F. Castro-Company, S. Romaguera, P. Tirado, A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application, J. Inequal. Appl. 2014 (122) (2014), 11 pp. [6] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994) 395–399. [7] V. Gregori, J.A. Mascarell, A. Sapena, On completion of fuzzy quasi-metric spaces, Topol. Appl. 153 (5–6) (2005) 886–899. [8] V. Gregori, S. Romaguera, Fuzzy quasi-metric spaces, Appl. Gen. Topol. 5 (1) (2004) 129–136. [9] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets Syst. 115 (2000) 485–489. [10] J. Gutiérrez-García, M.A. de Prada Vicente, Hutton [0, 1]-quasi-uniformities induced by fuzzy (quasi)-metric spaces, Fuzzy Sets Syst. 157 (6) (2006) 755–766. [11] J. Gutiérrez-García, S. Romaguera, M. Sanchis, Standard fuzzy uniform structures based on continuous t-norms, Fuzzy Sets Syst. 195 (2012) 75–89. [12] R.W. Heath, Some nonmetric, first countable, cancellative topological semigroups that are generalized metric spaces, Topol. Appl. 44 (1992) 167–173. [13] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logic – Studia Logica Library, vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.
JID:FSS AID:7238 /FLA
8
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.8 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
[14] I. Kramosil, J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975) 326–334. [15] J.-H. Kim, G.A. Anastassiou, Ch. Park, Additive ρ-functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl. 21 (6) (2016) 1115–1126. [16] K.Y. Lee, Approximation properties in fuzzy normed spaces, Fuzzy Sets Syst. 282 (2016) 115–130. [17] C. Liu, Metrizability of paratopological (semitopological) groups, Topol. Appl. 159 (2012) 1415–1420. [18] O.V. Ravsky, Paratopological groups I, Mat. Stud. 16 (2001) 37–48. [19] S. Romaguera, M. Sanchis, On fuzzy metric groups, Fuzzy Sets Syst. 124 (2001) 109–115. [20] I. Sánchez, M. Sanchis, Complete invariant fuzzy metrics on groups, Fuzzy Sets Syst. (2017), http://dx.doi.org/10.2016/j.fss.2016.12.019. [21] B. Schweizer, A. Sklar, Statistical metric spaces, Pac. J. Math. 10 (1960) 313–334.
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.1 (1-8)
Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
Fuzzy quasi-pseudometrics on algebraic structures Iván Sánchez ∗,1 , Manuel Sanchis 2 Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Spain Received 29 January 2017; received in revised form 29 April 2017; accepted 23 May 2017
Abstract In this work we state a number of theorems about fuzzy (quasi-)pseudometrizable algebraic structures. Our most useful results are: (1) a fuzzy semitopological group whose topology is induced by a left-invariant fuzzy (quasi-)pseudometric, then it is a fuzzy (paratopological) topological group, (2) if the topology on a semigroup S is induced by an invariant fuzzy quasi-pseudometric, then S is a fuzzy topological semigroup, and (3) the same conclusion is valid for a left-invariant fuzzy quasi-pseudometric on a monoid G such that the left translations are open and the right translations are continuous at the identity e of G. By means of the standard fuzzy (quasi-)pseudometric Md associated to a (quasi-)pseudometric d, our results apply in the case of semitopological groups, semigroups and monoids in order to obtain new results that allow us to generalize and to strengthen previous outcomes. © 2017 Elsevier B.V. All rights reserved.
Keywords: Fuzzy (quasi-)pseudometric; Left topological group; Right topological group; Semitopological group; Paratopological group; Topological group; Topological semigroup; Invariant fuzzy (quasi-)pseudometric
1. Introduction In this paper we shall focus our attention on topological algebraic structures equipped with a fuzzy quasipseudometric in the sense of Kramosil and Michalek. Combinations of a fuzzy metric structure and an algebraic structure deserve special attention in fuzzy Topological Algebra. The most frequently studied structures fall into the so-called fuzzy normed spaces (among others, the interested reader can consult [2,4,15,16]), although fuzzy metric topological groups are also worthy of consideration (see [11,19,20]). In [8] Gregori and Romaguera remove the symmetric condition in the definition of a fuzzy metric (in the sense of Kramosil and Michalek, see [14]) and introduce the notion of a fuzzy quasi-metric space. This allows us to consider nonsymmetric structures which fit in the realm of fuzzy nonsymmetric topology: fuzzy quasi-metric spaces and fuzzy quasi-normed spaces ([1,5,7,10]). In this context,
* Corresponding author.
E-mail addresses: [email protected] (I. Sánchez), [email protected] (M. Sanchis). 1 The author was supported by CONACYT of Mexico, grant number 259783. 2 The second author is supported by the Spanish Ministerio de Economía y Competitividad (Grant MTM2016-77143-P), Generalitat Valenciana
(Grant AICO/2016/030) and Universitat Jaume I (Grant P1-1B2014-35). http://dx.doi.org/10.1016/j.fss.2017.05.022 0165-0114/© 2017 Elsevier B.V. All rights reserved.
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.2 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
2
the classical results in semigroups and paratopological groups related to (left-)invariant metrics (see, for example, [17, Theorem 2.1], [18, Proposition 3.2]) encourage enough to merit further investigation in fuzzy nonsymmetric topology. In this paper we deal with one of the oldest problems in Topological Algebra: find sufficient conditions in order that a topological algebraic structure (in particular a nonsymmetric structure) become a stronger topological structure (in particular, a symmetric structure). Our goal is to obtain conditions on fuzzy quasi-metrics (respectively, fuzzy quasi-pseudometrics) semitopological groups and fuzzy paratopological groups (respectively, on fuzzy topological semigroups) which imply that they are either fuzzy paratopological groups or topological groups. To be precise, we show the following: (1) if (G, M, ∗) is a fuzzy semitopological group whose topology τM is induced by a left-invariant fuzzy (quasi-)pseudometric M, then (G, M, ∗) is a fuzzy (paratopological) topological group, (2) if (M, ∗) is an invariant fuzzy quasi-pseudometric on a semigroup S, then (S, M, ∗) is a fuzzy topological semigroup, and (3) the same conclusion is valid when (M, ∗) is a left-invariant fuzzy quasi-pseudometric on a monoid G such that the left translations are open and the right translations are continuous at the identity e of G. It is worth mentioning that, by means of the standard fuzzy (quasi-)pseudometric Md associated to a (quasi-)pseudometric d, our results apply in the case of semitopological groups, semigroups and monoids in order to obtain new results that allow us to generalize and to strengthen previous outcomes by Liu [17, Theorem 2.1] and Ravsky [18, Proposition 3.2]. A celebrated theorem by Birkhoff–Kakutani says that a Hausdorff topological group is metrizable if and only if it is first-countable. It is also a well-known result that a first-countable paratopological group is quasi-metrizable (see [18]). In spite of these results, a first-countable topological semigroup need not be quasi-pseudometrizable (see [12]). 2. Preliminaries According to [21], a continuous t-norm is a binary operation ∗ : [0, 1] × [0, 1] → [0, 1] which satisfies the following conditions: (i) ∗ is associative and commutative, (ii) ∗ is continuous, (iii) a ∗ 1 = a for every a ∈ [0, 1], and (iv) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, with a, b, c, d ∈ [0, 1]. It is a well-known fact, and easy to check, that for each continuous t-norm ∗ one has ∗ ≤ ∧, where ∧ is the continuous t-norm given by a ∧ b = min{a, b}. The interested reader is referred to [13] for further information on (continuous) t-norms. Following [8], a fuzzy quasi-pseudometric (in the sense of Kramosil and Michalek) 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 > 0: (i) (ii) (iii) (iv)
M(x, y, 0) = 0; M(x, x, t) = 1; M(x, z, t + s) ≥ M(x, y, t) ∗ M(y, z, s); M(x, y, _) : [0, +∞) → [0, 1] is left continuous.
A fuzzy pseudometric on X is a fuzzy quasi-pseudometric (M, ∗) on X which satisfies: (v) M(x, y, t) = M(y, x, t) for all x, y ∈ X and t > 0. By a fuzzy (quasi-)pseudometric space (in the sense of Kramosil and Michalek) we mean a triple (X, M, ∗) such that X is a set and (M, ∗) is a fuzzy (quasi-)pseudometric on X. Every fuzzy (quasi)-pseudometric (M, ∗) on a set X induces a topology τM on X having as a base the family {BM (x, ε, t) : x ∈ X, ε ∈ (0, 1), t > 0}, where BM (x, ε, t) = {y ∈ X : M(x, y, t) > 1 − ε} for all x ∈ X, ε ∈ (0, 1) and t > 0. Let (X, d) be a (quasi-)pseudometric space. Define a fuzzy set Md in X × X × [0, ∞) by ⎧ t ⎨ for all x, y ∈ X and t > 0; t + d(x, y) Md (x, y, t) = ⎩ 0 for all x, y ∈ X and t = 0. Then (Md , ∧) is a fuzzy (quasi-)pseudometric on X, and hence (Md , ∗) is a fuzzy quasi-pseudometric on X for all continuous t-norm ∗, the so-called (fixed a t-norm ∗ on X), the standard fuzzy (quasi-)pseudometric induced by d on X. It is known that the topology τM and the topology τd , induced by the (quasi-)pseudometric d, coincide (see [6,8,9]).
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.3 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
3
We now move on to notions from (fuzzy) Topological Algebra. Let G be an algebraic semigroup. Pick x ∈ G. The function λx : G → G defined by λx (g) = xg is called the left translation of G by x. Similarly, ρx : G → G defined as ρx (g) = gx is known as the right translation of G by x. A topological semigroup (G, τ ) is an algebraic semigroup G with a topology τ that makes the multiplication in G jointly continuous. A paratopological group G is a topological semigroup such that G is an algebraic group. We say that a paratopological group (G, τ ) is a topological group if the inverse is continuous, that is, if g −1 stands for the inverse element of g ∈ G, then the function g → g −1 from G onto G is continuous. (G, τ ) is said to be a left (respectively, right) topological group if the translations λx (respectively, ρx ) are continuous in (G, τM ) for all x ∈ G. The following result is a well known internal characterization of a (para)topological group. Theorem 2.1. ([3, Theorem 1.3.12]) Let G be a group with identity e and U a family of subsets of G containing e. If U satisfies the following conditions: (i) (ii) (iii) (iv)
for every U, V ∈ U, there exists W ∈ U such that W ⊆ U ∩ V ; for every U ∈ U and x ∈ U , there is V ∈ U such that V x ⊆ U ; for every U ∈ U and x ∈ G, we can find V ∈ U satisfying xV x −1 ⊆ U ; for every U ∈ U, there exists V ∈ U such that V 2 ⊆ U ;
then the family {U x : x ∈ G, U ∈ U} is a base for a topology τU on G. With this topology, G is a paratopological group, and the family {xU : x ∈ G, U ∈ U} is a base for the same topology on G. In addition, if U satisfies (v) for every U ∈ U, we can find V ∈ U with V −1 ⊆ U ; then (G, τU ) is a topological group. We will introduce the fuzzy structures we deal with Definition 2.2. By a fuzzy (quasi)-pseudometric semigroup we mean a triple (G, M, ∗) such that (G, M, ∗) is a fuzzy (quasi)-pseudometric space and (G, τM ) is a topological semigroup. A fuzzy (quasi)-pseudometric paratopological group is a fuzzy (quasi)-pseudometric semigroup (G, M, ∗) such that G is an algebraic group. We now address the last fuzzy structure we need: Definition 2.3. By a fuzzy (quasi)-pseudometric right topological group we mean a triple (G, M, ∗) such that (G, M, ∗) is a fuzzy (quasi)-pseudometric space and (G, τM ) is a right topological group. Fuzzy (quasi)-pseudometric left topological groups are defined in a similar way. If no confusion can arise, we write G for (G, τM ). The following notion plays a significant role in our results. Definition 2.4. A fuzzy quasi-pseudometric (M, ∗) on a semigroup G is left-invariant (respectively, right-invariant) if M(x, y, t) = M(ax, ay, t) (respectively, M(x, y, t) = M(xa, ya, t)) whenever a, x, y ∈ G and t > 0. We say that (M, ∗) is invariant if it is both left-invariant and right-invariant. Our notation and terminology are standard. For notions on Topological Algebra not defined here, the interested reader can consult [3]. 3. The results Our first result is concerning with the continuity of the operation on right fuzzy quasi-pseudometric topological groups. It provides a sufficient condition to obtain a fuzzy paratopological group. Theorem 3.1. If (G, M, ∗) is a fuzzy quasi-pseudometric right topological group such that (M, ∗) is left-invariant, then (G, M, ∗) is a fuzzy paratopological group.
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.4 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
4
Proof. Let e be the identity of G. According to [19], U = {B(e, n1 , n1 ) : n ∈ N} is a local base at e. Let us show that U = {B(e, n1 , n1 ) : n ∈ N} satisfies conditions (i)–(iv) in Theorem 2.1, that is, the topology τU associated to the family U makes G into a paratopological group. Item (i) follows from the fact that U is a local base at e in (G, τM ). To prove (ii) take n ∈ N and x ∈ B(e, n1 , n1 ). Since ρx is continuous at e and ρx (e) = x ∈ B(e, n1 , n1 ), there exists m ∈ N such that 1 1 1 1 1 1 , )) = B(e, , )x ⊆ B(e, , ). m m m m n n Thus, (ii) holds. Now, before to show (iii), we will prove that, for each n ∈ N and x ∈ G, we have: ρx (B(e,
1 1 1 1 xB(e, , ) = B(x, , ). n n n n
(1)
Indeed, take y ∈ B(e, n1 , n1 ). Since (M, ∗) is left-invariant, we have 1 1 1 M(x, xy, ) = M(e, y, ) > 1 − n n n so that xB(e, n1 , n1 ) ⊆ B(x, n1 , n1 ). For the other inclusion, pick z ∈ B(x, n1 , n1 ). Again, since (M, ∗) is left-invariant, we conclude that 1 1 1 M(e, x −1 z, ) = M(x, z, ) > 1 − . n n n This proves that x −1 z ∈ B(e, n1 , n1 ). Thus, z ∈ xB(e, n1 , n1 ) which shows (1). Let us show (iii). Pick n ∈ N and x ∈ G. Note that every right translation is a homeomorphism. So B(e, n1 , n1 )x is an open neighborhood of x. Hence there is m ∈ N such that B(x, m1 , m1 ) ⊆ B(e, n1 , n1 )x. This and (1) imply 1 1 −1 1 1 1 1 , )x = B(x, , )x −1 ⊆ B(e, , ). m m m m n n This proves (iii). Finally, we show (iv). Choose n ∈ N. Since the t-norm ∗ is continuous, there is k ∈ N such that for each q ≥ k we have xB(e,
(1 −
1 1 1 ) ∗ (1 − ) > 1 − . q q n
Put m = max{k, 2n}. Then, for each y, z ∈ B(e, m1 , m1 ), the following inequalities hold 1 2 1 1 M(e, yz, ) ≥ M(e, yz, ) ≥ M(e, y, ) ∗ M(y, yz, ) = n m m m 1 1 1 1 1 M(e, y, ) ∗ M(e, z, ) > (1 − ) ∗ (1 − ) > 1 − . m m m m n Therefore, B(e, m1 , m1 )B(e, m1 , m1 ) ⊆ B(e, n1 , n1 ). This proves (iv). By Theorem 2.1, (G, τU ) is a paratopological group and {xB(e, n1 , n1 ) : x ∈ G, n ∈ N} is a base for τU . Notice that equation (1) implies that {xB(e, n1 , n1 ) : x ∈ G, n ∈ N} also is a base for τM so that τU = τM . This shows that (G, M, ∗) is a fuzzy paratopological group. 2 If we replace fuzzy quasi-pseudometric by fuzzy pseudometric, we obtain a symmetric structure. Theorem 3.2. If (G, M, ∗) is a fuzzy pseudometric right topological group such that (M, ∗) is left-invariant, then (G, M, ∗) is a fuzzy topological group. Proof. By Theorem 3.1, (G, M, ∗) is a fuzzy paratopological group. We only need to prove that the family U = {B(e, n1 , n1 ) : n ∈ N} satisfies (v) of Theorem 2.1. Choose n ∈ N. If we take x ∈ B(e, n1 , n1 ), then
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.5 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
5
1 1 1 1 M(e, x −1 , ) = M(x, e, ) = M(e, x, ) > 1 − . n n n n We conclude that x −1 ∈ B(e, n1 , n1 ). So B(e, n1 , n1 ) is symmetric for every n ∈ N. It follows that (G, M, ∗) is a fuzzy topological group. 2 Arguing as in Theorems 3.1 and 3.2 we can obtain. Theorem 3.3. If (G, M, ∗) is a fuzzy quasi-pseudometric left topological group such that (M, ∗) is right-invariant, then (G, M, ∗) is a fuzzy paratopological group. If in addition, (G, M, ∗) is a fuzzy pseudometric left topological group, then (G, M, ∗) is a topological group. As far we know, the following four corollaries are new results on the classical theory. Corollary 3.4. Suppose that (G, τ ) is a left (right) topological group whose topology τ is induced by a right-(left-)invariant quasi-pseudometric. Then (G, τ ) is a paratopological group. Proof. Suppose that the topology τ is induced by a right-(left-)invariant quasi-pseudometric. Then the fuzzy quasipseudometric (Md , ∧) induced by d is right-(left-)invariant. Theorems 3.1 and 3.3 imply that (G, τM ) is a paratopological group. Since τM = τd = τ , we have that (G, τ ) is a paratopological group. 2 Remark 3.5. A topological group is balanced if it has a local base at the identity whose elements are invariant open sets. It is easy to see that if a fuzzy metric group G admits an invariant fuzzy metric, then G is balanced. The general linear group GL(2, R) is a fuzzy metric group, but it is not balanced (see [3, Exercise 1.8.d]). Hence GL(2, R) does not admit an invariant fuzzy metric. An argument similar to the one used in Corollary 3.4 allows us to obtain the following three results. Recall that a semitopological group G is a group G with a topology that makes the multiplication separately continuous. Note that a semitopological group is both a left and right topological group. Corollary 3.6. Suppose that (G, τ ) is a semitopological group whose topology τ is induced by a right-(or left-)invariant quasi-pseudometric d. Then (G, τ ) is a paratopological group. According to [17, Theorem 2.1], G is a metrizable topological group provided that G is a quasi-metrizable paratopological group with respect to a left continuous, left-invariant quasi-metric d where left-continuous means that the function d(g, _) is continuous for all g ∈ G. Notice that by Corollary 3.6 a semitopological group enjoying these properties is a paratopological group. Thus, we can extend [17, Theorem 2.1] in the following way: Corollary 3.7. Suppose that (G, τ ) is a semitopological group whose topology τ is induced by a left continuous, left-invariant quasi-metric. Then (G, τ ) is a metrizable topological group. Ravsky proved that if a paratopological group G is metrizable by a left-invariant metric, then G is a topological group (see [18, Proposition 3.2]). We now present two generalizations of this fact as a direct consequence of Theorems 3.1, 3.2 and 3.3. Corollary 3.8. Suppose that G is a left (right) topological group whose topology is induced by a right-(left-)invariant pseudometric. Then G is a topological group. It is easy to see that the topology on the Sorgenfrey line S is induced by the invariant quasi-metric d defined as d(x, y) = y − x if x ≤ y and d(x, y) = 1 if x > y. It is known that S is a paratopological group, but S is not a topological group. This shows that in Corollary 3.8, we cannot replace pseudometric by quasi-metric. Corollary 3.9. Suppose that (G, τ ) is a semitopological group whose topology τ is induced by a right-(or left-)invariant pseudometric d. Then (G, τ ) is a topological group.
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.6 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
6
We now move on to fuzzy semigroups. It is worth mentioning that Heath provides in [12] a valuable collection of nonmetrizable, (so, of nonfuzzy metrizable) first countable cancellative topological semigroups (for example, a nonmetrizable locally compact, cancellative abelian topological semigroup that is a quasimetrizable Moore space [12, Example 3.1]). Theorem 3.10. Let (M, ∗) be a fuzzy quasi-pseudometric on a semigroup S. If (M, ∗) is invariant, then (S, M, ∗) is a fuzzy topological semigroup. Proof. Take y, z ∈ S. Choose n ∈ N. Let k ∈ N be such that for each q ≥ k, (1 − q1 ) ∗ (1 − q1 ) > 1 − n1 . Put m = max{k, 2n}. If a ∈ B(y, m1 , m1 ) and b ∈ B(z, m1 , m1 ), then we have 1 2 1 1 M(yz, ab, ) ≥ M(yz, ab, ) ≥ M(yz, yb, ) ∗ M(yb, ab, ) = n m m m 1 1 1 1 1 M(z, b, ) ∗ M(y, a, ) ≥ (1 − ) ∗ (1 − ) > 1 − . m m m m n Therefore B(y, m1 , m1 )B(z, m1 , m1 ) ⊆ B(yz, n1 , n1 ). We have thus proved that multiplication is continuous in (G, τM ), i.e., (G, M, ∗) is a fuzzy topological semigroup. 2 Let us recall that a monoid is a semigroup with a neutral element. Theorem 3.11. Suppose that (M, ∗) is a left-invariant fuzzy quasi-pseudometric on a monoid G such that for each x ∈ G, λx is open and ρx is continuous at the identity e of (G, M, ∗). Then (G, M, ∗) is a fuzzy topological semigroup. Proof. Let e be the identity of G. We claim that for each n ∈ N and x ∈ G we have 1 1 1 1 xB(e, , ) ⊆ B(x, , ). n n n n
(2)
Indeed, take y ∈ B(e, n1 , n1 ). Since (M, ∗) is left-invariant, we have 1 1 1 M(x, xy, ) = M(e, y, ) > 1 − . n n n This proves (2). As a consequence of (2), we have that left translations are continuous at e. Now, we will show that for every n ∈ N, there is m ∈ N satisfying B(e,
1 1 1 1 1 1 , )B(e, , ) ⊆ B(e, , ). m m m m n n
(3)
For this, notice that, since the t -norm is continuous, there is k ∈ N such that for each q ≥ k we have that (1 − q1 ) ∗ (1 − q1 ) > 1 − n1 . Put m = max{k, 2n}. Then, for each y, z ∈ B(e, m1 , m1 ), the following inequalities hold: 1 2 1 1 M(e, yz, ) ≥ M(e, yz, ) ≥ M(e, y, ) ∗ M(y, yz, ) = n m m m 1 1 1 1 1 M(e, y, ) ∗ M(e, z, ) > (1 − ) ∗ (1 − ) > 1 − . m m m m n Therefore B(e, m1 , m1 )B(e, m1 , m1 ) ⊆ B(e, n1 , n1 ). Now, we will prove that the multiplication is continuous in (G, τM ). Take x, y ∈ G and n ∈ N. By (2), we have xyB(e, n1 , n1 ) ⊆ B(xy, n1 , n1 ). It follows from (3) that B(e, m1 , m1 )B(e, m1 , m1 ) ⊆ B(e, n1 , n1 ) for some m ∈ N. Hence xyB(e,
1 1 1 1 1 1 1 1 , )B(e, , ) ⊆ xyB(e, , ) ⊆ B(xy, , ). m m m m n n n n
(4)
By hypothesis, left translations are open. So yB(e, m1 , m1 ) is an open set in (G, τM ) which contains y. Also by hypothesis, ρy is continuous at e. Hence there is k ∈ N satisfying
JID:FSS AID:7238 /FLA
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.7 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
1 1 1 1 1 1 ρy (B(e, , )) = B(e, , )y ⊆ yB(e, , ). k k k k m m Therefore (4)–(5) imply
7
(5)
1 1 1 1 1 1 1 1 1 1 xB(e, , )yB(e, , ) ⊆ xyB(e, , )B(e, , ) ⊆ B(xy, , ). k k m m m m m m n n Since left translations are open, xB(e, 1k , 1k ) and yB(e, m1 , m1 ) are open neighborhoods of x and y, respectively. Therefore, multiplication in (G, τM ) is continuous. This completes the proof. 2 Applying the previous results, we get the following results in semigroups and topological monoids. Corollary 3.12. Suppose that d is a invariant quasi-pseudometric on a semigroup S. Then (S, d) is a topological semigroup. Question 3.13. Suppose that d is a left-invariant pseudometric on a semigroup S. Is (S, d) a topological semigroup? Corollary 3.14. Suppose that d is a left-invariant quasi-pseudometric on a monoid G such that for each x ∈ G, λx is open and ρx is continuous at the identity e of (G, d). Then (G, d) is a topological semigroup. 4. Conclusion We study how the existence of a (left-)invariant fuzzy (quasi-)pseudometric on an algebraic-topological structure (like a fuzzy left topological group or a semigroup) implies that actually we have a stronger structure (like a fuzzy (para)topological group). Our results fit in a long tradition of research in Topological Algebra. When applied to classical structures, they allow us to obtain new outcomes that generalizes helpful results of the theory. Moreover, since a fuzzy (quasi)uniformity is defined by a suitable family of fuzzy (quasi)pseudometrics, starting from the results presented in the paper, a future area of research is under what conditions on the natural fuzzy quasi-uniformities on a fuzzy topological semigroup (respectively, a fuzzy paratopological group) we can obtain a stronger fuzzy structure. Acknowledgements The authors wish to thank the referees for their valuable comments and suggestions for the improvement of this paper. References [1] C. Alegre, S. Romaguera, On the uniform boundedness theorem in fuzzy quasi-normed spaces, Fuzzy Sets Syst. 282 (2016) 143–153. [2] C. Alegre, S. Romaguera, The Hahn–Banach extension theorem for fuzzy normed spaces revisited, Abstr. Appl. Anal. (2014) 151472, 7 pp. [3] A.V. Arhangel’skii, M.G. Tkachenko, Topological Groups and Related Structures, Atlantis Ser. Math., vol. I, Atlantis Press and World Scientific, Paris–Amsterdam, 2008. [4] T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, Ann. Fuzzy Math. Inform. 6 (2) (2013) 271–283. [5] F. Castro-Company, S. Romaguera, P. Tirado, A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application, J. Inequal. Appl. 2014 (122) (2014), 11 pp. [6] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994) 395–399. [7] V. Gregori, J.A. Mascarell, A. Sapena, On completion of fuzzy quasi-metric spaces, Topol. Appl. 153 (5–6) (2005) 886–899. [8] V. Gregori, S. Romaguera, Fuzzy quasi-metric spaces, Appl. Gen. Topol. 5 (1) (2004) 129–136. [9] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets Syst. 115 (2000) 485–489. [10] J. Gutiérrez-García, M.A. de Prada Vicente, Hutton [0, 1]-quasi-uniformities induced by fuzzy (quasi)-metric spaces, Fuzzy Sets Syst. 157 (6) (2006) 755–766. [11] J. Gutiérrez-García, S. Romaguera, M. Sanchis, Standard fuzzy uniform structures based on continuous t-norms, Fuzzy Sets Syst. 195 (2012) 75–89. [12] R.W. Heath, Some nonmetric, first countable, cancellative topological semigroups that are generalized metric spaces, Topol. Appl. 44 (1992) 167–173. [13] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logic – Studia Logica Library, vol. 8, Kluwer Academic Publishers, Dordrecht, 2000.
JID:FSS AID:7238 /FLA
8
[m3SC+; v1.265; Prn:2/06/2017; 14:30] P.8 (1-8)
I. Sánchez, M. Sanchis / Fuzzy Sets and Systems ••• (••••) •••–•••
[14] I. Kramosil, J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975) 326–334. [15] J.-H. Kim, G.A. Anastassiou, Ch. Park, Additive ρ-functional inequalities in fuzzy normed spaces, J. Comput. Anal. Appl. 21 (6) (2016) 1115–1126. [16] K.Y. Lee, Approximation properties in fuzzy normed spaces, Fuzzy Sets Syst. 282 (2016) 115–130. [17] C. Liu, Metrizability of paratopological (semitopological) groups, Topol. Appl. 159 (2012) 1415–1420. [18] O.V. Ravsky, Paratopological groups I, Mat. Stud. 16 (2001) 37–48. [19] S. Romaguera, M. Sanchis, On fuzzy metric groups, Fuzzy Sets Syst. 124 (2001) 109–115. [20] I. Sánchez, M. Sanchis, Complete invariant fuzzy metrics on groups, Fuzzy Sets Syst. (2017), http://dx.doi.org/10.2016/j.fss.2016.12.019. [21] B. Schweizer, A. Sklar, Statistical metric spaces, Pac. J. Math. 10 (1960) 313–334.