Asymptotically Statistical Equivalent of Order α in Amenable Semigroups

In this study we introduce the concepts of asymptotically statistical equivalent functions of order α and strong asymptotically equivalent functions of order α defined on discrete countable amenable semigroups.

The idea of statistical convergence depends upon the density of subsets of the set N of natural numbers.The density of a subset E of N is defined by Marouf [19] introduced definitions for asymptotically equivalent sequences and asymptotic regular matrices.Patterson [25] extend these concepts by presenting an asymptotically statistically equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices.
Suppose that G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold, and w(G) and m(G) denote the spaces of all real valued functions and all bounded real functions on G, respectively.m(G) is a Banach space with the norm e-mail: yaltin23@yahoo.come-mail: mikailet68@gmail.come-mail: hifsialtinok@yahoo.comNomika [21] showed that, if G is a countable amenable group, there exists a sequence Here |A| denotes the number of elements the set A.

Main Results
Definition 1 Let α be any real number such that 0 < α ≤ 1.We define the upper and lower Folner α−density of the subset S of G by is called Folner α−density of S .For α = 1, we have Folner density of S ⊂ G which were defined an studied by Nuray and Rhoades ([22], [23], [24]).
Definition 2 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and let α be any real number such that 0 < α ≤ 1.The function f ∈ w(G) is said to be statistically convergent to s of order α for any Folner sequence The set of all statistically convergent functions of order α will be denoted by S α (G).
Definition 3 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold, α be any real number such that 0 < α ≤ 1.The function f ∈ w(G) is said to be strongly summable of order α to s for any Folner sequence {S n } for G if The set of all strongly summable functions of order α will be denoted by w α (G).
Definition 4 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and let α be any real number such that 0 < α ≤ 1.Two nonnegative functions f, h ∈ w(G) are said to be asymptotically statistical equivalent of order α, for any Folner sequence {S n } for G if, for every ε > 0, In this case we write f S α ∼ g.

Definition 5
Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and let α be any real number such that 0 < α ≤ 1.Two nonnegative functions f, h ∈ w(G) are said to be strong asymptotically equivalent of order α, for any Folner sequence {S n } for G if, for every ε > 0, In this case we write f Theorem 6 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and suppose also that the parameters α and β are fixed real numbers such Proof.Proof follows from the inequality From Theorem 6 we have the following.
Corollary 7 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and let α be any real number such that 0 < α ≤ 1, then f Theorem 8 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and suppose also that the parameters α and β are fixed real numbers such From Theorem 8 we have the following.
Corollary 9 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and let α be any real number such that 0 < α ≤ 1, then f Theorem 10 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and suppose also that the parameters α and β are fixed real numbers such From Theorem 10 we have the following results.
Corollary 11 Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and let α be any real number such that 0 < α ≤ 1, then f S α ∼ g implies f w α ∼ g.

Corollary 12
Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold and let α be any real number such that 0 < α ≤ 1, then f S α ∼ g implies f w ∼ g.

Corollary 13
Let G be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold, then f S ∼ g implies f w ∼ g.