ABSTRACT
The notion of “central subset” was first developed by H. Furstenberg [5] in the semigroup (N,+) of natural numbers, and then V. Bergelson and N. Hindman [1] defined *- central subset in any infinite discrete semigroup which is a natural extension of the prior one’s. V. Bergelson and N. Hindman developed the notion of central subset in any infinite discrete semigroup (G, +) (see Definition 2. 1), and pointed out any * -central subset of (G, +) is central in (G, T). Moreover, a result of Weiss (see [1, Theorem 6. ) guarantees that in a countable semigroup (G, +), a subset A ⊆ G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq3426.tif"/> is central in (G, +) if and only if A is * - central in (G, +). What about the uncountable semigroup (G,+)? Inspired by the result of Weiss, a necessary and sufficient condition concerning central subset in infinite (including countable and uncountable) discrete semigroup (G, +) is obtained in section 2. The notion of * * -central subset is defined (see Definition 2. 3), thus any central subset A of (G,) is *-central, or * * -central in (G, +).
