Hindman and Leader first introduced the notion of semigroup of ultrafilters converging to zero for a dense subsemigroups of $((0,\infty),+)$. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Tootkabani and Vahed generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent $e$ for a dense subsemigroups of a semitopological semigroup $(T, +)$ and they gave the combinatorial proof of central set theorem near $e$. Algebraically one can also define quasi-central sets near $e$ for dense subsemigroups of $(T, +)$. In a dense subsemigroup of $(T,+)$, C-sets near $e$ are the sets, which satisfy the conclusions of the central sets theorem near $e$. S. K. Patra gave dynamical characterizations of these combinatorially rich sets near zero. In this paper we shall prove these dynamical characterizations for these combinatorially rich sets near $e$.

中文翻译：

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幂等性附近的动力学

Hindman 和 Leader 首先引入了超滤波器半群的概念，对于 $((0,\infty),+)$ 的密集子半群收敛为零。使用 Stone-$\breve{C}$ech 紧化的代数结构，Tootkabani 和 Vahed 将这个概念推广并扩展到幂等而不是零，即超滤子的半群收敛到密集子半群的幂等 $e$的半拓扑半群 $(T, +)$ 并且他们给出了 $e$ 附近的中心集合定理的组合证明。对于$(T, +)$ 的密集子半群，代数上还可以定义$e$ 附近的准中心集。在$(T,+)$的稠密子半群中，$e$附近的C-sets是满足$e$附近中心集定理结论的集合。SK Patra 给出了这些接近于零的组合丰富集合的动态特征。