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In this paper, the concepts of σ -uniform density of subsets A of the setN of positive integers and corresponding Iσ convergence of functions defined on discrete countable amen bl semigroups were introduced. Furthermore, for any Foln er sequence inclusion relations between Iσ -convergence and invariant convergence also Iσ -convergence and [Vσ ]p-convergence were given. We introduce the concept of Iσ -statistical convergence and Iσ -lacunary statistical convergence of functions defined on d iscrete countable amenable semigroups. In addition to these definitions, we gi ve some inclusion theorems. Also, we make a new approach to th e notions of [V,λ ]-summability,σ -convergence andλ -statistical convergence of Folner sequences by using idea ls and introduce new notions, namely,Iσ -[V,λ ]-summability,Iσ -λ -statistical convergence of Folner sequences. We mainly ex amine the relation between these two methods as also the relation between Iσ -statistical convergence and Iσ -λ -statistical convergence of Folner sequences introduced b y the author recently.

On I_{?}-convergence of folner sequence on amenable semigroups