ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 11 Issue 1(2019), Pages 28-35.

SOME GENERALIZED CONVERGENCE TYPES USING IDEALS IN AMENABLE SEMIGROUPS

U ˘GUR ULUSU, ERDINC¸ D ¨UNDAR, FATIH NURAY

Abstract. The aim of this study is to introduce the concepts ofI-statistically

convergence,I-statistically pre-Cauchy sequence andI-stronglyp-summability for functions defined on discrete countable amenable semigroups and to exam-ine some properties of these concepts.

1. Introduction and Background

The concept of statistical convergence was introduced by Fast [7] and this concept has been studied by ˇSal´at [16], Fridy [8], Connor [1] and many others. The idea of

I-convergence which is a generalization of statistical convergence was introduced by Kostyrko et al. [9] which is based on the structure of the ideal I of subset of the set natural numbersN. Then, Das et al. [2] introduced a new notion, namely

I-statistical convergence by using ideal. Recently, Das and Sava¸s [3] introduced the notion ofI-statistically pre-Cauchy sequence.

The concepts of summability in amenable semigroups were studied in [5, 6, 10, 11]. In [13], Nuray and Rhoades introduced the notions of convergence and sta-tistical convergence in amenable semigroups. Also, the notion of almost stasta-tistical convergence in amenable semigroups studied by Nuray and Rhoades [14]. Fur-thermore, the concepts of asymptotically statistical equivalent functions defined on amenable semigroups investigated by Nuray and Rhoades [14].

The aim of this study is to introduce the concepts ofI-statistically convergence,

I-statistically pre-Cauchy sequence andI-stronglyp-summability for functions de-fined on discrete countable amenable semigroups and to examine some properties of these concepts. For the particular case; when the amenable semigroup is the additive positive integers, our definitions and theorems yield the results of [2, 3].

LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold, andw(G) and m(G) denote the spaces of all real valued functions and all bounded real functions onG, respectively. m(G) is a Banach space with the supremum normkfk∞= sup{|f(g)|:g∈G}. Namioka [12] showed that, if Gis a countable amenable group, there exists a sequence{Sn} of finite subsets ofGsuch that

2010Mathematics Subject Classification. 40A05, 40A35, 43A07.

Key words and phrases. Statistical convergence; Ideal convergence; Pre-Cauchy sequence; Fol-ner sequence; Amenable semigroups.

c

2019 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.

Submitted December 19, 2018. Published February 21, 2019. Communicated by F. Basar.

i) G=S∞_{n}_{=1}Sn,

ii) Sn⊂Sn+1 (n= 1,2, ...),

iii) lim n→∞

|Sng∩Sn|

|Sn| = 1, nlim→∞

|gSn∩Sn|

|Sn| = 1,

for allg∈G, where |A|denotes the number of elements the finite setA.

Any sequence of finite subsets ofGsatisfying (i),(ii) and (iii) is called a Folner sequence forG.

The sequenceSn={0,1,2, ..., n−1}is a familiar Folner sequence giving rise to the classical Ces`aro method of summability.

Now, we recall the basic definitions and concepts (See, [2, 8, 9, 13]). A sequencex= (xk) is statistically convergent toL if for everyε >0

lim n→∞

1

n

k≤n:|xk−L| ≥ε

= 0.

A family of setsI ⊆2N_{is called an ideal if and only if}

(i)∅ ∈ I, (ii) For each A, B∈ I we haveA∪B∈ I, (iii) For eachA∈ I and eachB⊆Awe haveB ∈ I.

An ideal is called non-trivial ifN∈ I/ and non-trivial ideal is called admissible if

{n} ∈ I for eachn∈N. All ideals in this paper are assumed to be admissible. A sequencex= (xk) is said to beI-convergent toLif for everyε >0

A(ε) =

k∈N:|xk−L| ≥ε ∈ I.

A sequence x = (xk) is said to be I-statistically convergent to L if for every

ε >0 andδ >0

n∈_{N}: 1

n

k≤n:|xk−L| ≥ε

≥δ

∈ I.

A sequence x= (xk) is said to beI-statistical pre-Cauchy if for anyε >0 and

δ >0

n∈_{N}: 1

n2

(j, k) :|xj−xk| ≥ε, j, k≤n ≥δ

∈ I.

LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. A functionf ∈w(G) is said to be convergent to s for any Folner sequence {Sn} forG if, for everyε > 0 there exists ak0 ∈N such that|f(g)−s|< ε, for allm > k0 andg∈G\Sm.

LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. A functionf ∈w(G) is said to be a Cauchy sequence for any Folner sequence{Sn}forGif, for everyε >0 there exists ak0∈N such that|f(g)−f(h)|< ε, for allm > k0andg, h∈G\Sm.

LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. A function f ∈w(G) is said to be strongly summable tosfor any Folner sequence{Sn}forGif

lim n→∞

1

|Sn|

X

g∈Sn

|f(g)−s|= 0.

LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and 0< p <∞. A function f ∈w(G) is said to be stronglyp-summable tosfor any Folner sequence{Sn}forGif

lim n→∞

1

|Sn|

X

g∈Sn

The upper and lower Folner densities of a setS⊂Gare defined by

δ(S) = lim sup n→∞

1

|Sn|

{g∈Sn :g∈S}

and

δ(S) = lim inf n→∞

1

|Sn|

{g∈Sn:g∈S}

,

respectively. Ifδ(S) =δ(S), then

δ(S) = lim n→∞

1

|Sn|

{g∈Sn:g∈S}

,

is called Folner density ofS. TakeG=N,Sn={0,1,2, ..., n−1} andS be the set
of positive integers with leading digit 1 in the decimal expansion. The setS has
no Folner density with respect to the Folner sequence {Sn}, since δ(S) = 1_{9} and

δ(S) = 5 9.

LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. A functionf ∈w(G) is said to be statistically convergent tosfor any Folner sequence{Sn}forGif, for everyε >0

lim n→∞

1

|Sn|

g∈Sn:|f(g)−s| ≥ε = 0.

LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. A functionf ∈w(G) is said to be a statistically Cauchy sequence for any Folner sequence{Sn}forGif, for everyε >0 andm≥0, there exists anh∈G\Smsuch that

lim n→∞

1

|Sn|

g∈Sn:|f(g)−f(h)| ≥ε = 0.

2. _{Main Results}

In this section, we introduced the concepts of I-statistically convergence, I -statistically pre-Cauchy sequence and I-strongly p-summability for functions de-fined on discrete countable amenable semigroups and examined some properties of these concepts.

Definition 2.1. LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈w(G)is said to beI-statistical convergent tos, for any Folner sequence{Sn} forG, if for every ε, δ >0

n∈_{N}: 1

|Sn|

g∈Sn :|f(g)−s| ≥ε ≥δ

∈ I.

In this case we writef −→SI s.

The set of allI-statistical convergent functions will be denotedSI(G).

Theorem 2.2. TheI-statistical convergence off ∈w(G)depends on the particular choice of the Folner sequence.

Example 2.3. Let G=Z2 and take two Folner sequences as follows:

{S_{n}1}=

(i, j)∈Z2:|i| ≤n,|j| ≤n and {Sn2}=

(i, j)∈Z2:|i| ≤n,|j| ≤n2

and define f(g)∈w(G)by

f =

1 , if (i, j)∈A,

0 , if (i, j)∈/A.

where

A=(i, j)∈Z2:i≤j ≤n, i= 0,1,2, ..., n; n= 1,2, ... .

Since for the Folner sequence{S2

n}

lim n→∞

1

|S2

n|

g∈S2_{n}:|f(g)−0| ≥ε = lim

n→∞

(n+1)(n+2) 2

(2n+ 1)(2n2_{+ 1)} = 0,

i.e.,

n∈_{N}: 1

|S2

n|

g∈S_{n}2:|f(g)−0| ≥ε ≥δ

∈ Id,

thenf(g)isI-statistically convergent to0. But, since for the Folner sequence{S1

n}

lim n→∞

1

|S1

n|

g∈S_{n}1:|f(g)−0| ≥ε = lim

n→∞

(n+1)(n+2) 2

(2n+ 1)2 6= 0,

i.e.,

n∈N: 1

|S1

n|

g∈Sn2:|f(g)−0| ≥ε ≥δ

/ ∈ Id,

thenf(g)is notI-statistically convergent to0.

Definition 2.4. LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈w(G)is said to beI-statistical pre-Cauchy, for any Folner sequence {Sn} forG, if for every ε, δ >0 such that

n∈N: 1

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥ε ≥δ

∈ I.

Theorem 2.5. If f ∈w(G)is I-statistically convergent for Folner sequence {Sn} forG, then it isI-statistically pre-Cauchy for same sequence.

Proof. Letf ∈w(G) beI-statistical convergent tosfor Folner sequence {Sn} for

G. Then for everyε, δ >0, we have

Aε,δ=

n∈_{N}: 1

|Sn|

n

g∈Sn:|f(g)−s| ≥

ε

2

o ≥δ

∈ I.

Hence, for alln∈Ac_{ε,δ}where cstands for the complement, we get

1

|Sn|

n

g∈Sn :|f(g)−s| ≥

ε

2

o < δ,

i.e.,

1

|Sn|

n

g∈Sn:|f(g)−s| ≥

ε

2

o

>1−δ,

Now, taking

Bε=

n

g∈Sn :|f(g)−s|<

ε

2

we observe that forg, h∈Bε

|f(g)−f(h)| ≤ |f(g)−s|+|f(h)−s|< ε

2 +

ε

2 =ε. Then

Bε×Bε⊂

(g, h)∈Sn:|f(g)−f(h)|< ε which implies that

_{|}_{B}

ε|

|Sn|

2 ≤ 1

|Sn|2

(g, h)∈Sn :|f(g)−f(h)|< ε .

Thus, for alln∈Ac ε,δ

1

|Sn|2

(g, h)∈Sn :|f(g)−f(h)|< ε ≥
_{|}_{B}

ε|

|Sn|

2

>(1−δ)2,

i.e.,

1

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥ε <1−(1−δ)2.

Letµ >0 be given. Choosing δ >0 so that 1−(1−δ)2 _{< µ}_{, we see that for all}
n∈Ac

ε,δ

1

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥ε < µ

and so

n∈N: 1

|Sn|2

(g, h)∈Sn :|f(g)−f(h)| ≥ε ≥µ

⊂Aε,δ.

SinceAε,δ∈ I, so

n∈N: 1

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥ε ≥µ

∈ I.

Hence,f is I-statistical pre-Cauchy sequence. _{}

Theorem 2.6. f ∈ m(G) is I-statistically pre-Cauchy for Folner sequence {Sn} forGif and only if for everyε >0,

(

n∈_{N}: 1

|Sn|2

X

g,h∈Sn

|f(g)−f(h)| ≥ε )

∈ I.

Proof. Firstly suppose that for everyε >0,

( n∈N:

1

|Sn|2

X

g,h∈Sn

|f(g)−f(h)| ≥ε )

∈ I.

Note that for anyε >0 andn∈_{N}, we have

1

|Sn|2

X

g,h∈Sn

|f(g)−f(h)| ≥ε
_{1}

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥ε

Hence, for anyδ >0

n∈N:

1

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥ε ≥δ

⊂ (

n∈N: 1

|Sn|2

X

g,h∈Sn

|f(g)−f(h)| ≥δε )

Due to our acceptance, the set on the right hand side belongs toI which implies that

n∈_{N}: 1

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥ε ≥δ

∈ I.

This shows thatf isI-statistically pre-Cauchy.

Conversely, suppose that f ∈ m(G) is I-statistically pre-Cauchy. Since f ∈ m(G), setkfk∞=M. Letδ >0 be given. Then, for everyn∈Nwe can write

1

|Sn|2

X

g,h∈Sn

|f(g)−f(h)| ≤ ε

2 + 2M

_{1}

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥

ε

2

.

Sincef isI-statistically pre-Cauchy, for everyδ >0

Aε,δ=

n∈N: 1

|Sn|2

(g, h)∈Sn :|f(g)−f(h)| ≥

ε

2 ≥δ

∈ I.

Hence, for alln∈Ac_{ε,δ}, we get

1

|Sn|2

(g, h)∈Sn:|f(g)−f(h)| ≥

ε

2 < δ

and so

1

|Sn|2

X

g,h∈Sn

|f(g)−f(h)| ≤ ε

2 + 2M δ.

Letµ >0 be given. Then, choosingε, δ >0 so that ε_{2}+ 2M δ < µwe observe that
for alln∈Ac_{ε,δ},

1

|Sn|2

X

g,h∈Sn

|f(g)−f(h)|< µ,

i.e.,

n n∈N:

1

|Sn|2

X

g,h∈Sn

|f(g)−f(h)| ≥µo⊂Aε,δ∈ I.

This completed the proof. _{}

Definition 2.7. LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈w(G)is said to be I-summable tos, for any Folner sequence{Sn} forG, if for every ε >0

( n∈N:

1

|Sn|

X

g∈Sn

f(g)−s

≥ε )

∈ I.

In this case we writef −→CI s.

The set of allI-summable functions will be denotedCI(G).

Definition 2.8. LetGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈w(G) is said to be I-strongly summable tos, for any Folner sequence{Sn} forG, if for every ε >0

(

n∈N: 1

|Sn|

X

g∈Sn

|f(g)−s| ≥ε )

∈ I.

The set of allI-strongly summable functions will be denotedCI[G].

Definition 2.9. Let0< p <∞andGbe a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈w(G)is said to beI-stronglyp-summable tos, for any Folner sequence {Sn}forG, if for every

ε >0

( n∈N:

1

|Sn|

X

g∈Sn

|f(g)−s|p≥ε )

∈ I.

In this case we writef [C]

p I

−→s.

The set of allI-stronglyp-summable functions will be denotedC_{I}p[G].

Theorem 2.10. Let 0 < p < ∞. If f ∈ w(G) is I-strongly p-summable to s for Folner sequence {Sn} forG, then it isI-statistically convergent to s for same sequence.

Proof. For anyf ∈w(G), fix anε >0. Then, we can write

X

g∈Sn

|f(g)−s|p _{=} X

g∈Sn

|f(g)−s|≥ε

|f(g)−s|p_{+} X

g∈Sn

|f(g)−s|<ε

|f(g)−s|p

≥

g∈Sn:|f(g)−s| ≥ε εp

and so

1

εp _{|}_{S}
n|

X

g∈Sn

|f(g)−s|p_{≥} 1

|Sn|

g∈Sn:|f(g)−s| ≥ε .

Hence, for anyδ >0

n∈_{N}: 1

|Sn|

g∈Sn:|f(g)−s| ≥ε ≥δ

⊆ (

n∈N: 1

|Sn|

X

g∈Sn

|f(g)−s|p_{≥}_{δ ε}p

) .

Therefore, due to our acceptance, the right set belongs toI, so we get

n∈N: 1

|Sn|

g∈Sn :|f(g)−s| ≥ε ≥δ

∈ I.

This proof completed. _{}

Theorem 2.11. Let 0 < p <∞. If f ∈ m(G) is I-statistically convergent to s for Folner sequence {Sn} for G, then it is I-strongly p-summable to s for same sequence.

Proof. Let f ∈ m(G) be I-statistically convergent to s for Folner sequence {Sn} forG. Sincef ∈m(G), set kfk∞+s=M. Letε >0 be given. Then we have

1

|Sn|

X

g∈Sn

|f(g)−s|p _{=} 1

|Sn|

X

g∈Sn

|f(g)−s|≥ε

2

|f(g)−s|p_{+} 1

|Sn|

X

g∈Sn

|f(g)−s|<ε

2

|f(g)−s|p

≤ M

p

|Sn|

n

g∈Sn:|f(g)−s| ≥

ε

2

o +

ε

2

and so

( n∈N:

1

|Sn|

X

g∈Sn

|f(g)−s|p_{≥}_{ε}

)

⊆

n∈N: 1

|Sn|

n

g∈Sn:|f(g)−s| ≥

ε

2

o ≥

ε

2Mp

.

Therefore, due to our acceptance, the right set belongs toI, so we get

(

n∈N: 1

|Sn|

X

g∈Sn

|f(g)−s|p_{≥}_{ε}

) ∈ I.

This proof completed. _{}

Acknowledgments. The authors would like to thank the anonymous referee for his/her comments.

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U˘gur Ulusu, Erdinc¸ D¨undar, Fatih Nuray

Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe Uni-versity, 03200, Afyonkarahisar, Turkey

E-mail address:ulusu@aku.edu.tr

E-mail address:edundar@aku.edu.tr