PII. S0161171204307052 http://ijmms.hindawi.com © Hindawi Publishing Corp.
T
-NEIGHBORHOOD GROUPS
T. M. G. AHSANULLAH and FAWZI AL-THUKAIR
Received 8 July 2003
We generalize min-neighborhood groups to arbitraryT-neighborhood groups, whereTis a continuous triangular norm. In particular, we point out that our results accommodate the previous theory on min-neighborhood groups due to T. M. G. Ahsanullah. We show that everyT-neighborhood group isT-uniformizable, therefore,T-completely regular. We also present several results of T-neighborhood groups in conjunction withT I-groups due to J. N. Mordeson.
2000 Mathematics Subject Classification: 54A40, 54H11.
1. Introduction. Menger in [19] introduces an important class of T-uniformities (T being a continuoust-norm) that is generated by a probabilistic metric [21]. Moti-vated by the Menger’sT-uniformities, Höhle [13] brought into light in his celebrated article the idea of probabilistic metrization of fuzzy uniformities. While developing his theory, he showed that a fuzzy T-uniformity is probabilistic metrizable if and only if it is Hausdorff-separated and has a countable base. He also pointed out that when T =min is considered, his fuzzyT-uniformity reduces to min-fuzzy uniformity of R. Lowen—a fuzzy uniformity widely used over the years. One of the interesting features of this min-fuzzy uniformity [16] is that it gives rise to a fuzzy neighborhood space [17]; an interesting and very well-behaved class of fuzzy topological spaces [15], used by many authors in a wide variety of ways. Among the prominent classes of so called fuzzy neighborhood spaces are, for instance, Katsaras linear fuzzy neighborhood spaces [14], fuzzy metric neighborhood spaces [18], fuzzy neighborhood groups, rings, modules, algebras, and commutative division rings [1,2,3,4,5].
Very recently, following the famous articles of Menger [19], Höhle [13], Frank [9], Hashem and Morsi [10,11,12] introduced a class of fuzzy topological spaces [15] as they put it: fuzzyT-neighborhood spaces herein calledT-neighborhood spaces—a nat-ural generalization of min-fuzzy neighborhood spaces of Lowen [17]. Our main target here in this article is to generalize the notion of mfuzzy neighborhood groups in-troduced in [2]. We show that everyT-neighborhood group is aT-uniform space, and therefore, aT-complete regular space in the sense of Hashem and Morsi [12]. We also generalize the two important characterization theorems, which give necessary and suf-ficient condition for aT-neighborhood system and a group structure to be compatible, and a prefilter to be aT-neighborhood prefilter.
2. Preliminaries. LetT be a continuous two-place function (known as continuous triangular norm ort-norm) mapping from the closed unit square to the closed unit in-terval satisfying certain conditions. In other words,T:I×I→I,(α,β)αT β, satisfying the following conditions:
(Ta) 0T0=0,αT1=αfor allα∈I; (Tb) αT β=βT αfor all(α,β)∈I×I;
(Tc) ifα≤βandγ≤δ, thenαT γ≤βT δfor allα,β,γ,δ∈I; (Td) (αT β)T γ=αT (βT γ)for allα,β,γ∈I.
Definition2.1[6,16]. A nonempty subsetᏮ⊂IG is called a prefilterbase if and only if the following conditions are true:
(PB1) 0∈Ꮾ;
(PB2) for allν1,ν2∈Ꮾ, there existsν∈Ꮾsuch thatν≤ν1∧ν2.
IfᏮis a prefilterbase inIG, then by its saturation we understand the following
col-lection:
Ꮾ∼=ν:G →I; ∀ >0∃ν
∈Ꮾν−≤ν. (2.1)
Definition2.2[10,11,12]. AT-neighborhood space is anI-topological space [15] (G,−)whose closure operator “−” is induced by some indexed familyΩ=(Ω(x))
x∈G
of prefilterbases inIGdefined by
¯
ξ(x)= inf
ν∈Ω(x)supz∈Gξ(z)T ν(z) ∀ξ∈I
G, x∈G. (2.2)
Theorem 2.3[10, 11, 12]. A familyΩ=(Ω(x))x∈G of prefilterbases in IG is aT
-neighborhood base in G if and only if it satisfies the following two properties for all
x∈G:
(TB1) for allν∈Ω(x),ν(x)=1;
(TB2) for allν∈Ω(x), there exists a family(νy∈Ω(y))(y,)∈G×I0 which satisfies for all(y,)∈G×I0,
sup
z∈G
νx,(z)T νz,(y)≤ν(y)+. (2.3)
The familyΩis said to be a T-neighborhood basis for(G,−), and everyν∈Ω(x) is calledT-neighborhood at x. ThisI-topology is denoted by tT(Ω). However, from
now on we will be calling the triple(G,−,t(Ω))the T-neighborhood space withΩa T-neighborhood base inG.
Theorem 2.4[10, 11,12]. Let(G1,−,t(Ω1))and(G2,−,t(Ω2))be T-neighborhood
spaces withT-neighborhood basesΩ1andΩ2, respectively. Then a functionf:G1→G2 is continuous atx∈G1
⇐⇒ ∀µ∈Ω2
f (x), f−1(µ)∈Ω 1(x)∼,
⇐⇒ ∀µ2∈Ω2f (x) ∀ >0∃µ1∈Ω1(x)µ1−≤f−1µ2, ⇐⇒f−1(σ )(x)≤f−1(σ )¯ (x) ∀σ∈IG2.
IfΛ,Γ∈IG×G andν∈IG, thenT-section ofΛoverνis given by
ΛνT(x)=sup
y∈Gν(y)TΛ(y,x) ∀x∈G. (2.5)
TheT-composition ofΛandΓ is defined as
Λ◦TΓ(x,y)=sup z∈G
Γ(x,z)TΛ(z,y) ∀(x,y)∈G×G. (2.6)
Γ is called symmetric ifΓs=Γ, that is,Γ(y,x)=Γ(x,y), for all(x,y)∈G×G.
Definition2.5[10,11,12]. A subsetᏮ⊂IG×Gis called aT-uniform base on a set Gif and only if the following properties are fulfilled:
(TUB1) Ꮾis a prefilterbase;
(TUB2) for allx∈G, for allν∈Ꮾ,ν(x,x)=1;
(TUB3) for allβ∈Ꮾ, for all >0, there existsβ∈Ꮾsuch thatβ◦Tβ−≤β;
(TUB4) for allβ∈Ꮾ, for all >0, there existsβ∈Ꮾsuch thatβ−≤β.
The collectionᏮof fuzzy subsets ofG×Gis calledT-quasi-uniform base on a set G if and only if it fulfills the preceding conditions (TUB1), (TUB2), and (TUB3), while ᐁis calledT-quasi-uniformity if and only if Ꮾ =ᐁ. AT-uniformityᐁis a saturated T-uniform baseᏮ.
Theorem 2.6[10, 11, 12]. IfᏮ is aT-quasi-uniform base on a setG, then for all x∈G, the family
Σ(x)=β1x |β∈Ꮾ∼=β1x |β∈Ꮾ∼ (2.7)
is aT-neighborhood system onG.
Proposition2.7[10,11,12]. Let(G,ᐁ)be aT-quasi-uniform space. Then the closure
of theT-neighborhood space(G,t(ᐁ))is given by
¯ µ=inf
σ∈ᐁσµT ∀µ∈I
G. (2.8)
Theorem2.8[10,11,12]. If(G,τ)is a topological space andᐂτ=(ᐂτ(x))x∈Gis its
associated neighborhood system inG, then(G,−,t(Ω
τ)), a generated topological space, generated byτ, is aT-neighborhood space with aT-neighborhood basisΩ=(Ω(x))x∈G, where for allx∈G,
Ω1:=Ω1(x)=1M:G →I;M∈ᐂτ(x)⊂IG;
Ω2:=Ω2(x)=
1M:G →I;x∈M∈τ⊂IG;
Ω3:=Ω3(x)=
ν:G →I; νis l.s.c. in x andν(x)=1⊂IG.
(2.9)
Just for the sake of convenience, we provide the proof of the following proposition.
Proposition2.9. A functionf:(G,ᐂτ)→(G,ᐂτ)between two topological spaces
is continuous at a pointx∈Gif and only iff:(G,t(Ωτ))→(G,t(Ωτ))is continuous at
Proof. Letf :(G,ᐂτ)→(G,ᐂτ) be continuous atx∈G andµ∈Ω
τ(f (x)); in
view ofTheorem 2.4, we show thatf−1(µ)∈Ω
τ(x).
ChooseM∈ᐂ
τ(f (x))such that µ=1M. This implies that there exists anM ∈
ᐂτ(x)such thatf (M)⊂M, and hence for all >0,
1M(x)−=1−≤1f−1(M)(x)=f−1µ. (2.10)
Withµ=1M, one obtainsµ−≤f−1(µ)implying thatf−1(µ)∈Ωτ(x).
Conversely, we show that the functionf:(G,ᐂτ)→(G,ᐂτ)is continuous atx∈G.
If U∈ᐂ
τ(f (x)), then 1U ∈Ωτ(f (x)) impliesf−1(1U)∈Ωτ(x)by continuity off
between the generated spaces.
Thus, for all >0, there is aµ=µ∈Ωτ(x)such that
µ−≤f−11
U. (2.11)
This implies that for all >0, there exists aV∈ᐂτ(x)such that 1V=µ=µand
1V−≤1f−1(U). (2.12)
NowV∈ᐂτ(x)impliesx∈Vif and only if 1V(x)=1. Therefore,
0<1−=1V(x)−≤1f−1(U)(x) ⇒1f−1(U)(x) >0⇒1f−1(U)(x)
=1⇐⇒x∈f−1(U). (2.13)
This means thatV⊆f−1(U)impliesf−1(U)∈ᐂτ(x). That is,fis continuous atx∈G.
Theorem2.10[11]. Let(G,−)be anI-topological space. Then(G,−)is a T
-neigh-borhood space if and only ifαT µ=αTµ¯for allµ∈IGand for allα∈I.
Definition2.11[12]. AnI-topological space(X,τ)is calledT-completely regular ifτ is the initialI-topology for the family of all continuous functions from(X,τ)to (Ᏸ+,tT(Ᏺ
Ᏼ)).
Here,Ᏸ+ stands for the collection of all distance distribution functions from+to the unit intervalI, and the pair(Ᏸ+,tT(ᏲᏴ))is theT-neighborhood space induced by
the well-known Höhle’s probabilisticT-metricᏲᏴ. For details, we refer to [12,13].
Theorem2.12[12]. T-complete regularity is equivalent toT-uniformizability.
3. Some results onT-neighborhood spaces
Theorem 3.1. Let(G1,−,t(Ω1)) and (G2,−,t(Ω2)) be twoT-neighborhood spaces
with basesΩ1=(Ω1(x))x∈G1 andΩ2=(Ω2(x))x∈G2 inG1andG2, respectively. Then their T-product(G1×G2,−⊗T,t(Ω1)⊗Tt(Ω2)) is theT-neighborhood space with base Ω=Ω1⊗TΩ2defined by
Ω(x,y)=ν1⊗Tν2|ν1∈Ω1(x), ν2∈Ω2(y)
whereν1⊗Tν2is given as
ν1⊗Tν2(x,y)=ν1(x)T ν2(y) ∀(x,y)∈G1×G2. (3.2)
Moreover,
ν1⊗Tν2=ν1⊗Tν2 ∀ν1∈IG1, ν2∈IG2. (3.3)
Conversely, if
ν1⊗Tν2=ν1⊗Tν2 ∀ν1∈IG1, ν2∈IG2, (3.4)
then both theI-topological spaces(G1,−)and(G2,−)areT-neighborhood spaces.
Proof. First we show that for all(x,y)∈G1×G2,Ω(x,y)is a prefilterbase. (PB1) Obviously,Ω≠∅and 0∈Ω.
(PB2) Letξ1,ξ2∈Ω(x,y), then there areν1,ν2∈Ω1(x)andµ1,µ2∈Ω2(y)such that
ξ1=ν1⊗Tµ1andξ2=ν2⊗Tµ2.
Now,ξ1∧ξ2=(ν1⊗Tµ1)∧(ν2⊗Tµ2). For any(x,y)∈G1×G2,
ξ1(x,y)∧ξ2(x,y)=ν1⊗Tµ1(x,y)∧ν2⊗Tµ2(x,y) =ν1(x)T µ1(y)
∧ν2(x)T µ2(y)
≥ν1(x)∧ν2(x)
Tν1(y)∧µ2(y)
=ν1∧ν2
(x)Tµ1∧µ2
(y).
(3.5)
Therefore,ξ1∧ξ2(x,y)≥ν(x)∧µ(y)for some ν∈Ω1(x)andµ ∈Ω2(y), since
bothΩ1(x)andΩ2(y)are prefilterbases inG1andG2, respectively.
This implies thatξ1∧ξ2(x,y)≥ν⊗Tµ(x,y)=ξ(x,y)andξ∈Ω(x,y), and hence
ξ≤ξ1∧ξ2, proving thatΩ(x,y)is a prefilterbase inG1×G2.
Now we prove the conditions ofTheorem 2.3.
(TB1) If x∈G and ξ∈Ω(x,x), then for some ν ∈Ω1(x)and µ ∈Ω2(x), we have
ξ(x,x)=ν⊗Tµ(x,x),ν(x)T µ(x)=1T1=1.
(TB2) Let(x,y)∈G1×G2,ξ∈Ω(x,y), and∈I0. Then there existsν∈Ω1(x)and
µ∈Ω2(y)such thatξ=ν⊗Tµ.
Consequently, there is a family(νy1∈Ω1(y1))(y1,)∈G1×I0, aT-kernel for ν which
satisfies for all(y1,)∈G1×I0,
sup
z1∈G1
νx,z1
T νz1,y1
≤νy1
+. (3.6)
Also, there is a family(µy2∈Ω2(y2))(y2,)∈G2×I0, aT-kernel forµwhich satisfies for
all(y2,)∈G2×I0,
sup
z2∈G2
Now for all(y1,y2)∈G1×G2,
ν⊗Tµy1,y2
+δ=νy1
T µy2
+δ≥νy1
+Tµy2
+ (3.8)
with=T ,δ>0.
This yields that
ν⊗Tµy1,y2+δ ≥ sup
z1∈G1
νx,z1
T νz1,y1
T sup
z2∈G2
µy,z2
T µz2,y2
= sup
(z1,z2)∈G1×G2
νx,z1T µy,z2Tνz1,y1T µz2,y2
= sup
(z1,z2)∈G1×G2
νx,⊗Tµy,z1,z2
Tνz1,⊗Tµz2,y1,y2
⇒ sup
(z1,z2)∈G1×G2
νx,⊗Tµy,z1,z2
T νz1,⊗Tµz2,y1,y2
≤ν⊗Tµy1,y2+δ.
(3.9)
In order to prove the final part, we proceed as follows. Let ν1∈IG1, ν2∈IG2, and
(x,y)∈G1×G2.
Then in view ofDefinition 2.2, we have
ν1⊗Tν2(x,y)= inf
ξ1∈Ω1(x)ξ2∈infΩ2(y)z1sup∈G1z2sup∈G2
ν1⊗Tν2
z1,z2
Tξ1⊗Tξ2
z1,z2
= inf
ξ1∈Ω1(x)ξ2∈infΩ2(y)z1sup∈G1z2sup∈G2
ν1
z1
T ν2
z2
Tξ1
z1
T ξ2
z2
= inf
ξ1∈Ω1(x)ξ2∈infΩ2(y)z1sup∈G1z2sup∈G2ν1
z1T ξ1z1T ν2z2T ξ2z2
= inf
ξ1∈Ω1(x)z1sup∈G1ν1
z1
T ξ1
z2
T inf
ξ2∈Ω2(y)z2sup∈G2ν2
z2
T ξ2
z2
=ν1(x)T ν2(y)=ν1⊗Tν2(x,y).
(3.10) To prove the converse part, we proceed as follows. Since
ν1⊗Tν2=ν1⊗Tν2 ∀ν1∈IG1, ν2∈IG2, (3.11)
in view ofTheorem 2.10, we have
αT ν1⊗Tν2
(x)=(αT ν1⊗Tν2
(x)
=α(x)Tν1⊗Tν2(x) =α(x)Tν1(x)T ν2(x).
(3.12)
Since this holds for allxand for allν1andν2, withν2=1, we have
αT ν1⊗Tν2(x)=(αT ν1⊗Tν2(x)=α(x)Tν1(x)T1 =α(x)T ν1(x)=
αT ν1
(x)=αT ν1
so (G1,−)is aT-neighborhood space. Similarly, withν1=1, we see that(G2,−)is a
T-neighborhood space. This completes the proof.
Proposition 3.2. Let (G1,−,t(Ω1)) and (G2,−,t(Ω2)) be T-neighborhood spaces.
Then the projections
pr1:
G1×G2,−⊗T,tΩ1⊗TtΩ2 →G1,−,tΩ1, x1,x2→x1,
pr2:
G1×G2,−⊗T,t
Ω1
⊗TtΩ2
→G2,−,t
Ω2
, x1,x2
→x2,
(3.14)
are continuous.
Proof. Letν∈Ω1(x1)and >0. Then
pr−1 1
ν1x1,x2=ν1pr1
x1,x2=ν1x1T1≥ν1x1T ν2x2≥ν1⊗Tν2x1,x2−
⇒ν1⊗Tν2−≤pr−11
ν1 ⇒pr−11
ν1∈Ωx1,x2∼.
(3.15)
This implies that pr1:(G1×G2,−⊗T,t(Ω1)⊗Tt(Ω2))→(G1,−,t(Ω1)),(x1,x2)x1, is
continuous, and similarly, one can prove that pr2:(G1×G2,−⊗T,t(Ω1)⊗Tt(Ω2))→
(G2,−,t(Ω2)),(x1,x2)x2, is continuous.
Definition 3.3. A T-neighborhood space (G,−,t(Ω))is said to be a T N-regular space if and only if for all z∈G, for all µ ∈Ω(z), and for all >0, there exists a ν∈Ω(z)closed such that
+µ(z)≥ inf
ρ∈Ω(z)supt∈Gν(t)T ρ(t)
=ν(z)¯ . (3.16)
Theorem3.4. EveryT-quasi-uniform space(G,Ψ)isT N-regular.
Proof. Suppose thatz∈G,ψ∈Ψ, and >0, and chooseψ∈Ψsuch that
ψ◦Tψ≤ψ+. (3.17)
Ift∈G, then by usingProposition 2.7,
ψzT(t)= inf
ψ∈Ψsupy∈Gψz(y)T ψ
(y,t)≤sup
y∈Gψ(z,y)T ψ(y,t)
=ψ◦Tψ(z,t)≤ψ(z,t)+ =ψz(t)+.
(3.18)
Hence the result follows.
4. T-neighborhood groups. In what follows, we consider(G,·)as a multiplicative group witheas the identity element. Ifµ:G→I, thenµ−1(x)is defined asµ−1(x)=
Definition4.1. Let(G,·)be a group and(G,−,t(Ω))aT-neighborhood space with T-neighborhood baseΩonG. Then the quadruple(G,·,−,t(Ω))is called aT- neighbor-hood group if and only if the following properties are satisfied:
(TG1) the mappingm:(G×G,−⊗T,t(Ω)⊗
Tt(Ω))→(G,−,t(Ω)), (x,y)xy, is
con-tinuous;
(TG2) the inversion mappingr:(G,−,t(Ω))→(G,−,t(Ω)),xx−1, is continuous.
A group structure and aT-neighborhood system is said to be compatible if and only if (TG1) and (TG2) are fulfilled.
Remarks4.2. AT-neighborhood group may not be a fuzzy topological group in the sense of Foster [8] since we have usedT-neighborhood topology, which differ from the product fuzzy topology.
Proposition4.3. Let(G,·)be a group and(G,−,t(Ω))aT-neighborhood space with
aT-neighborhood baseΩ. Then the quadruple(G,·,−,t(Ω))is aT-neighborhood group
if and only if the mapping
h:G×G,−⊗T,t(Ω)⊗
Tt(Ω)→G,−,t(Ω), (x,y)→xy−1, (4.1)
is continuous.
Proof. Observe that the conditions (TG1) and (TG2) are equivalent to the following
single condition:
(TG3) the mappingh:(G×G,−⊗T,t(Ω)⊗
Tt(Ω))→(G,−,t(Ω)),(x,y)xy−1, is
con-tinuous.
In fact, if we let f (x,y)=(x,y−1), then by (TG2),f is continuous and hence in
conjunction with (TG1), one obtains the continuity ofh. On the other hand, (TG3)⇒(TG2) for x→ex−1 =x−1 is then continuous; while (TG1) follows from (TG3) and (TG2),
because(x,y)→x(y−1)−1=xyis then continuous.
Proposition4.4. Let(G,·)be a group and(G,−,t(Ω))aT-neighborhood space with ΩaT-neighborhood base inG. Then
(a) the mappingm:(G×G,·,−⊗T,t(Ω)⊗Tt(Ω))→(G,·,−,t(Ω)), (x,y)xy, is continuous at(e,e)∈G×Gif and only if for allµ∈Ω(e), for all >0, there exists
ν∈Ω(e)such that
νTν≤µ+; (4.2)
(b) the inversion mappingr:(G,·,−,t(Ω))→(G,·,−,t(Ω)),xx−1, is continuous at
e∈Gif and only if for allµ∈Ω(e), for all >0, there existsν∈Ω(e)such that
ν≤µ−1+. (4.3)
Proof. (a) In view ofTheorem 2.4, continuity at(e,e)∈G×Gis equivalent to
∀µ∈Ω(e)⇒m−1(µ)∈Ω(e)⊗
TΩ(e)∼
⇐⇒ ∀µ∈Ω(e), ∀ >0, ∃ν=ν∈Ω(e)ν⊗Tν≤m−1(µ)+.
Butm(ν⊗Tν)(z)=sup(x,y)∈m−1(z)ν(x)T ν(y)=supxy=zν(x)T ν(y)=νTν(z). Thus,
in this case, continuity at(e,e)∈G×Gis in fact equivalent to
νTν≤µ+. (4.5)
(b) This follows almost in the same way as in (a).
Corollary4.5. If(G,·,−,t(Ω))is aT-neighborhood group, then the mapping (4.1)
is continuous at(e,e)∈G×Gif and only if for allµ∈Ω(e)and for all >0, there exists aν∈Ω(e)such that
νTν−1≤µ+. (4.6)
Proof. This follows at once from the composition of (a) and (b) inProposition 4.4.
Proposition 4.6. Let (G,−,t(Ω)) be a T-neighborhood space and A ⊂ G. Then (A,−,t(Ω|A)) is a T-neighborhood space, a subspace of the T-neighborhood space (G,−,t(Ω)).
Proof. The proof follows by easy verification.
Theorem4.7. The triple(G,·,τ)is a topological group if and only if the quadruple (G,·,−,t(Ω
τ)), where Ωis the generated T-neighborhood basis, is aT-neighborhood group.
Proof. With the help ofProposition 2.9, it follows that the mapping
h:G×G,ᐂτ×ᐂτ →G,ᐂτ, (x,y)→xy−1, (4.7)
is continuous if and only if
h:G×G,tΩτ⊗TtΩτ →G,tΩτ, (x,y)→xy−1, (4.8)
is continuous, whereΩis the basis for the generatedT-neighborhood spaces.
Lemma4.8. Let(G,·,−,t(Ω))be aT-neighborhood group anda∈G. Then
(1) the left translationᏸa:(G,·,−,t(Ω))→(G,·,−,t(Ω)),xax, and the right trans-lationa:(G,·,−,t(Ω))→(G,·,−,t(Ω)),xax, are homeomorphisms;
(2) the inner automorphismᏵa:(G,·,−,t(Ω))→(G,·,−,t(Ω)),zaza−1, is an iso-morphism;
(3) ν∈Ω(e)∼if and only ifᏸa(ν)∈Ω(a)∼ if and only ifa(ν)∈Ω(a)∼. In other words, ifΩis saturated, thenν∈Ω(e)if and only if1{a}Tν=aTν∈Ω(a)if and only ifνTa∈Ω(a);
(4) ν∈Ω(a)∼if and only ifᏸ−a(ν)∈Ω(e)∼if and only if−a(ν)∈Ω(e)∼. In other words, ifΩis saturated, thenν∈Ω(a)if and only if1{a−1}Tν=a−1ν∈Ω(e) if and only ifνTa−1∈Ω(e);
(5) ifν∈Ω(e), thenν−1∈Ω(e);
Proof. (1) follows at once from the definitions while (2) follows from the fact that Ᏽa=ᏸa◦−a=−a◦ᏸafor alla∈G.
(3) Letν∈Ω(e)∼⊂Ω(e), that is,ν∈Ω(a−1a)=Ω(ᏸ−1
a (a)). Sinceᏸ−a1is continuous,
then in view ofTheorem 2.4, ᏸa(ν)=(ᏸ−a1)−1(ν)∈Ω(a)∼ impliesᏸa(ν)∈Ω(a)∼.
Conversely, letᏸa(ν)∈Ω(a)∼⊂Ω(a)impliesᏸa(ν)∈Ω(a)impliesᏸa(ν)∈Ω(a)= Ω(ae)=Ω(ᏸa(e)), and sinceᏸa:G→Gis continuous injection again byTheorem 2.4,
ν=ᏸ−1
a (ᏸa(ν))∈Ω(e)∼. For the calculations of the other part, see [7, Theorem 5.1.1].
(4) follows from (3) while (5) follows from the fact that the inversion mappingr:G→
G,xx−1is a homeomorphism.
(6) We haveνTν−1=(νTν−1)−1. Ifx∈G, then
νTν−1−1(x)=νTν−1x−1= sup
ab=x−1ν(a)T ν
b−1
= sup
st−1=x−1ν(s)T ν(t)=tssup−1=xν(t)T ν(s) = sup
ts−1=xν(t)T ν
s−1−1
= sup
ts−1=xν(t)T ν
−1s−1
=νTν−1(x).
(4.9)
This completes the proof.
Definition4.9. AT-neighborhood space(G,−,t(Ω))is called homogeneous space if and only if for all(a,b)∈G×G, there exists a homeomorphismf :(G,−,t(Ω))→ (G,−,t(Ω))such thatf (a)=b.
Theorem4.10. EveryT-neighborhood group is a homogeneous space.
Proof. This follows from the fact that for alla,b∈G×G, the function
a−1b:G →G, x→xa−1b, (4.10)
is a homeomorphism.
Lemma4.11. Let(G,·)be a group, and let, for all µ∈IG,µL:G×G→I,(x,y) µL(x,y)=µ(x−1y)(resp.,µR:G×G→I,(x,y)µR(x,y)=µ(yx−1)) be the vicinities
L-associated (resp.,R-associated) withµ.
Then for allµ,θ,ν∈IG,(x,y)∈G×G, and triangular normT:I×I→I, the following hold:
(1) µLθT=θTµ(resp.,µRθT=µTθ);
(2) µLT νL=(µT ν)L(resp.,µRT νR=(µT ν)R);
(3) (µs
L)=(µL)s;
(4) µLTνL=(νTµ)L(resp.,µRTνR=(νTµ)R).
Proof. (1) For all(x,θ,µ)∈G×IG×IG,
µLθT(x)=sup y∈G
θ(y)T µL(y,x)=sup y∈G
θ(y)T µy−1x
=θTµ(x) by [7, Theorem 5.1.1].
(2) and (3) are obvious. (4) For all(x,y)∈G×G,
µLTνL(x,y)=sup z∈G
νL(x,z)T µL(z,y)=sup z∈G
νx−1zT µz−1y
= sup
st=x−1zz−1y=x−1y
ν(s)T µ(t)=νTµx−1y
=νTµL(x,y).
(4.12)
Theorem4.12. EveryT-neighborhood group is aT-uniform space.
Proof. If(G,·,−,t(Ω))is aT-neighborhood group, then(G,−,t(Ω))is aT -neighbor-hood space with theT-neighborhood basisΩ.
We consider the following collection:
Ω=µL|µ∈Ω(e)⊂IG×G. (4.13)
We claim thatΩis aT-uniform basis. (TUB1) ClearlyΩis a prefilterbasis.
(TUB2) Ifψ∈Ω, then there exists aµ∈Ω(e)such thatψ=µL, and for allx∈G,
ψ(x,x)=µL(x,x)=µ(e)=1. (4.14)
(TUB3) Ifψ∈Ω, then there exists aµ∈Ω(e)such thatψ=µL.
Thus, by virtue ofProposition 4.4(a), for all >0, we can findν∈Ω(e)such that
ν
Tν−≤µ. (4.15)
If we letν
L=ψ, then one obtains
ψTψ−=νLTνL−=νTνL−≤µL
⇒ψTψ−≤ψ. (4.16)
(TUB4) If ψ ∈ Ω, then there is a µ ∈ Ω(e) such that ψ = µL. Consequently, by Proposition 4.4(b), for all >0, there exists aν∈Ω(e)such that
ν−≤µ−1. (4.17)
Therefore,νL−≤(µ−1)
L=(µL)−1impliesψ−≤ψs.
This shows in accordance withDefinition 2.5thatΩis aT-uniform basis, which in turn gives rise to a leftT-uniformityᐁL=Ω∼.
In fact, we have for allx∈G,
ᐁL(x)=µL1x |µ∈Ω(e)∼=ᏸx(µ)|µ∈Ω(e)∼=Ω(x)∼, (4.18)
which is a T-neighborhood system onG and that(G,t(Ω)=t(ᐁL)) is a T-uniform
space. Similarly, one can obtain rightT-uniformity.
Proof. This follows from the preceding theorem in conjunction withTheorem 2.12
because everyT-neighborhood group is T-uniformizable and everyT-uniformizable space isT-completely regular.
Theorem4.14. Let(G,·)be a group,(G,−,t(Ω))aT-neighborhood space withT
-neighborhood baseΩinG. Then the quadruple(G,·,−,t(Ω))is aT-neighborhood group
if and only if the following are true:
(1) for alla∈G,Ω(a)∼= {ᏸa(µ)|µ∈Ω(e)}∼(resp., for alla∈G,Ω(a)∼= {a(µ)|
µ∈Ω(e)}∼);
(2) for allµ∈Ω(e), for all >0, there exists aν∈Ω(e)such that
νTν≤µ+, (4.19)
that is, the mappingm:(x,y)xyis continuous at(e,e)∈G×G;
(3) for allµ∈Ω(e), for all >0, there exists aν∈Ω(e)such that
ν≤µ−1+, (4.20)
that is, the mappingr:xx−1is continuous ate∈G;
(4) for allµ∈Ω(e), for all >0, and for alla∈Gsuch that
aTνTa−1≤µ+, (4.21)
that is, the mappingᏵa:xaxa−1is continuous ate∈G.
Proof. Let(G,·,−,t(Ω))be aT-neighborhood group. Then the conditions (1), (2), (3), and (4) are clearly true.
To prove the converse part, we remark that fromCorollary 4.5, it follows that the mappingh:G×G→G;(x,y)xy−1is continuous at(e,e), and since the translations
ᏸa andaare continuous ataande, respectively, the continuity ofmfollows from
the following chain:
G×G ᏸa−1×ᏸb−1
→G×G m→G Ᏽb→G ᏸab−1
→G, (4.22)
where(a,b)→(e,e)→e→e→ab−1.
Theorem4.15. Let(G,·)be a group andᏲ a collection of nonempty subsets ofIG,
that is,∅ =Ᏺ⊂IGsuch that
(1) Ᏺis a prefilterbasis andµ(e)=1for allµ∈Ᏺ;
(2) for allµ∈Ᏺ, for all >0, there exists aν∈Ᏺsuch thatν−≤µ−1;
(3) for allµ∈Ᏺ, for all >0, there exists aν∈Ᏺsuch thatνTν−≤µ;
(4) for allµ∈Ᏺ, for alla∈G, for all >0, there existsν∈Ᏺsuch thataTνT
a−1−≤µ.
Then there exists a uniqueT-neighborhood system compatible with the group structure ofGsuch thatᏲis aT-neighborhood basis ate∈G.
We let
Ꮾ=µL|µ∈Ᏺ⊂IG×G. (4.23)
We show thatᏮis aT-quasi-uniform basis for aT-quasi-uniformity. We verifyDefinition 2.5upto (TUB3).
(TUB1) Ꮾis a prefilterbasis; for 0∈Ꮾwhich is clearly true, sinceᏲis a prefilterbasis. Next, letλ,ξ∈Ꮾ, thenλ=µLfor someµ∈Ᏺandξ=ηLfor someη∈Ᏺ. SinceᏲ
is a prefilterbasis, there exists aθ∈Ᏺsuch thatθ≤µ∧ηandθL≤µL∧ηL=λ∧ξ,
proving thatᏮis indeed a prefilterbasis.
(TUB2) For all x ∈ G, and ψ ∈Ꮾ, we have ψ =µL for some µ ∈ Ᏺ and ψ(x,x) =
µL(x,x)=µ(e)=1 by (1).
(TUB3) Letψ∈Ꮾ. Then there exists aµ∈Ᏺsuch thatψ=µL.
Now by (3), for all >0, we can find aν∈Ᏺsuch that
νTν−≤µ. (4.24)
But then by virtue ofLemma 4.11(4), we getνLTνL−≤µL. So, if we putψ=νL, then
ψTψ−≤ψ. (4.25)
This completes the proof thatᏮis aT-quasi-uniform basis which in turn gives rise to aT-quasi-uniformity and hence aT-quasi-uniform space. Then in view of theTheorem 2.6, since everyT-quasi-uniform space is aT-neighborhood space, in this case, we have theT-neighborhood system as given by the family
µL1x T|µL∈Ꮾ∼=µL1x T|µL∈Ꮾ∼ =1xTµ|µ∈Ᏺ∼ =1xµ|µ∈Ᏺ∼.
(4.26)
Thus one obtains the T-neighborhood system with the following family: Ω(x)= {1xTµ|µ∈Ᏺ}, a basis for the system in question.
Theorem4.16. Let(G,·,−,t(Ω))be aT-neighborhood group. Then for allµ:G→I,
¯
µ=infµTν|ν∈Ω(e)∼=infµTν|ν∈Ω(e)∼. (4.27)
Proof. Observe that everyT-neighborhood group is aT-quasi-uniform space. Therefore, by virtue ofTheorem 2.6, we can write, in particular, that
¯
µ=infνLµT|ν∈Ω(e)∼. (4.28)
Then by usingLemma 4.11(1), we have the following:
¯
Corollary4.17. In aT-neighborhood group(G,·,−,t(Ω)), the following property
holds:
¯
µ=infνTµ|ν∈Ω(e)∼=infνTµ|µ∈Ω(e)∼. (4.30)
Proof. This follows at once from the preceding results.
Theorem 4.18. If (G,·,−,t(Ω))is aT-neighborhood group, then(G,−,t(Ω))is T
-regular.
Proof. Letµ∈Ω(e)and >0. Since the map(x,y)xy−1is continuous at(e,e)∈ G×G, in view ofCorollary 4.5, we can find aν∈Ω(e)such that
νTν−1≤µ+. (4.31)
Then usingTheorem 4.16, we obtain
¯
µ(x)= inf
ω∈Ω(e)νTω
−1≤ν
Tν−1≤µ(x)+, (4.32)
which ends the proof.
Proposition4.19. If(G,·,−,t(Ω))is aT-neighborhood group, then for allµ,ν∈IG,
we have the following:
(i) ¯µTν¯≤µTν;
(ii) µ−1=µ¯−1;
(iii) xTµTy=xTµ¯Tyfor allx,y∈G.
Proof. (i) Ifz∈G, then we have
¯
µTν(z)¯ = sup
xy=zµ(x)T¯ νy¯ =(x,y)sup∈m−1(z)
¯
µ⊗Tν¯(x,y)
=mµ¯⊗Tν¯(z)=mµ⊗Tν(z)
≤m[µ⊗Tν](z)=µTν(z).
(4.33)
(ii) and (iii) follow immediately.
Lemma4.20. If(G,·)and(G,·)are groups andf:G→Gis a group homomorphism,
then
fxTa−1Tµ=f (x)Tf (a)−1Tf (µ). (4.34)
Proof. This follows the same way as in [2, Lemma 2.15]; see also [7].
Theorem4.21. Let(G,·,−,t(Ω))and(H,·,−,t(Ξ))beT-neighborhood groups with
basesΩandΞinGandH, respectively. Iff:G→His a group homomorphism, thenf
is continuous if and only if it is continuous at one point.
Proof. Letf:G→Hbe continuous at the pointa∈G. We need to show thatf is continuous at eachx∈G. Letξ∈Ξ(f (x))and >0. Then we havef (x)−1
Tξ∈Ξ(e)
pointa∈Gyields thatf−1(f (a)
Tf (x)−1Tξ)∈Ω(a)∼, which in turn implies that
there exists aσ =σ∈Ω(a)such that
σ−≤f−1f (a)
Tf (x)−1Tξ. (4.35)
Now we haveµ:=xTa−1Tσ∈Ω(x).
Thus, one obtains
µ(z)−=xTa−1Tσ (z)−=σax−1z− ≤f (a)Tf (x)−1Tξfax−1z =fax−1
Tξfax−1f (z) =ξf (z)=f−1(ξ)(z),
(4.36)
that is,µ−≤f−1(ξ), which implies thatf−1(ξ)∈Ω(x)∼.
Now we present some results onT-neighborhood groups in conjunction with Morde-son’sT I-group.
5. Application ofT-neighborhood groups inT I-groups
Definition5.1[7,20]. AnI-subsetµofGis called aT I-subgroup ofGif it fulfills the following conditions:
(G1) µ(e)=1;
(G2) µ(x−1)≥µ(x), for allx∈G;
(G3) µ(xy)≥µ(x)T µ(y), for allx,y∈G.
We denote the set of allT I-subgroups ofGbyT I(G)and that of the set of all normal T I-subgroups byNT I-subgroups, while byNI-subgroup we mean normalI-subgroups, the one introduced by Rosenfeld [20] in which caseT=min is used.
Proposition5.2. Let(G,·,−,t(Ω))be aT-neighborhood group andµ∈T I(G). Then ¯
µt(Ω)∈T I(G).
Proof. In view of [7, Theorem 5.1.4], it suffices to prove that
¯ µt(Ω)
Tν¯t(Ω)−1≤µ¯t(Ω). (5.1)
Sinceµ∈T I(G), we haveµTµ−1≤µ. Then by an easy calculation, one obtains
µTµ−1=hµ⊗Tµ−1, (5.2)
which in conjunction withTheorem 3.1, yields the following:
¯ µt(Ω)
Tµ¯t(Ω)−1=hµ¯t(Ω)⊗Tµ¯t(Ω)−1
≤µ¯t(Ω), (5.3)
Proposition5.3. If(G,·,−,t(Ω))is aT-neighborhood group andµ∈NT I(G), then ¯
µt(Ω)∈NT I(G).
Proof. Sinceµ ∈NI(G), we have Ix(µ)=xTµTx−1 =µ, where Ix :G→G, zxzx−1is an inner automorphism. But then usingProposition 4.19(iii), we obtain
xTµ¯Tx−1=xTµx−1=µ.¯ (5.4)
Hence the result follows from [7, Theorem 5.2.1(N5)].
Acknowledgments. We express our sincere gratitude to the anonymous referees
for scrupulous checking and providing us with their kind suggestions. This work is supported by the College of Science, King Saud University (KSU) Research Center, Project no. MATH/1423/07(2003/07).
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T. M. G. Ahsanullah: Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia
E-mail address:[email protected]
Fawzi Al-Thukair: Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia
