Central sets theorem on noncommutative semigroups (Q2312699)

In this paper, the authors formulate a new version of Central Sets Theorem and generalize it to arbitrary semigroups. Some algebraic properties are derived. Let \((S,+)\) be a commutative infinite semigroup, \(A\subseteq S\) and \(T\) be a nonempty set. Then \(A\) is a \(J_T\)-\textit{set} if and only if whenever \(F\in P_f(^T S)\), there exist \(a\in \mathcal{S}\) and \(H\in P_f(T)\) such that for each \(f\in F\), \(a+\sum\limits_{t\in H}f(t)\in A\). Here \(P_f(T)\) denotes the collection of all nonempty subsets of \(T\) and \(^T S\) the collection of all functions from \(T\) into \(S\). If \(A\) is a \(J_\mathbb{N}\)-set then it is a \(J_S\)-set. The reverse is true if \(S\) is countable. But if \(S\) is uncountable and \(A\) is a \(J_S\)-set, is \(A\) a \(J_\mathbb{N}\)-set? \noindent \(\bullet\) If \(A\) is a piecewise syndetic subset (then it is \(J_\mathbb{N}\)-set) of a commutative semigroup \(S\), then \(A\) is a \(J_S\)-set. (\(A\subseteq S\) is \textit{piecewise syndetic} if and only if there exists some \(G\in P_f(S)\) such that \(\{-a+(\bigcup\limits_{t\in G}(-t+A)):a\in S\}\) has the finite intersection property.) Let \((S,+)\) be an infinite discrete semigroup. The collection of all ultrafilters on \(S\) is called the Stone-\(\check{\text{C}}\)ech \textit{compactification} of \(S\) and denoted by \(\beta S\). There is a unique extension of the operation to \(\beta S\) such that \((\beta S,+)\) is a right topological semigroup. Set \(K(\beta S)\) denotes the union of all minimal left (right) ideals of \(\beta S\). An element \(p\in \beta S\) is an \textit{\textit{idempotent}} if and only if \(p+p=p\). A subset \(A\) of \(S\) is called \textit{central} if and only if there is some idempotent \(p\in K(\beta S)\) such that \(A\in p\). In the new version of Central Sets Theorem set \(^\mathbb{N} S\) is replaced by \(^S S\). \noindent \(\bullet\) Let \(A\) be a central subset of an infinite commutative semigroup \(S\). There exist two functions \(\alpha: P_f(^S S)\rightarrow S\) and \(H: P_f(^S S)\rightarrow P_f(S)\) such that \noindent (1) if \(F,G\in P_f(^S S)\) and \(F\subsetneq G\), then \(H(F)\cap H(G)=\emptyset\), and \noindent (2) whenever \(G_1,\ldots, G_m\in P_f(^S S)\), \(G_1\subsetneq \cdots \subsetneq G_m\), and for each \(i\in\{1,\ldots,m\}\), \(f_i\in G_i\), one has \(\sum\limits_{i=1}^m (\alpha(G_i)+\sum\limits_{t\in H(G_i)}f_i(t))\in A\). If in the thesis of above theorem we change \(^S S\) by \(^T S\), where \(T\) is a nonempty set, the subset \(A\) is called \(C_T\)-\textit{set}. If \(A\) a is a \(C_\mathbb{N}\)-set then it is a \(C_S\)-set. The reverse is true if \(S\) is countable. But if \(S\) is uncountable and \(A\) is a \(C_S\)-set, then is \(A\) a \(C_\mathbb{N}\)-set? \noindent\(\bullet\) Let \(A\) be a \(C_S\)-set in a infinite commutative semigroup \(S\), and \((f_i)_{i=1}^{\infty}\) be a sequence in \(^S S\). There exist a sequence \((a_n)_{n=1}^{\infty}\) in \(S\) and a sequence \((H_n)_{n=1}^{\infty}\) in \(P_f(S)\) such that \(H_n\cap H_m=\emptyset\) for each \(m\neq n\) such that for each \(u\in\Phi:=\{f\in ^{\mathbb{N}}\mathbb{N}:f(n)\leq n\,\,\forall n\in \mathbb{N}\}\), \(FS((a_n+\sum\limits_{t\in H_n}f_{u(n)}(t))_{n=1}^{\infty})\subseteq A\). (Here \(FS\) denotes finite sums.) In particular, the conclusion applies if \(A\) is a central set in \(S\). The authors continue this process for non commutative semigroups. In fact, \(J_S\)-sets and \(C_S\)-sets are defined with respect the collection \(^S S\). For a free semigroup \(S\) on an alphabet with al lest two members, the collection of all \(J_S\)-sets have the partition regular property. So this guaranties that \(J_S(S):=\{p\in\beta S: \forall A\in p,\,\, A\text{ is a }J_S-\text{set}\}\neq\emptyset\). Also \(J_S(S)\) is a compact two sided ideal of \(\beta S\). The Central Theorem for arbitrary semigroups is the following. \noindent \(\bullet\) Let \(S\) be a semigroup, let \(A\) be a central subset of \(S\), and let \((f_i)_{i=1}^{\infty}\) be a sequence in \(^S S\). Given \(m\in \mathbb{N}\), \(a\in S^{m+1}\), and \(H\in \mathcal{J}_m:=\{(H_1,\ldots, H_m)\in P_f(S)^m:\text{if }m>1\text{ and }1\leq i\neq j\leq m, \text{ then } H_i\cap H_j=\emptyset\},\) then there exist sequences \((m(n))_{n=1}^{\infty}\), \((a_n)_{n=1}^{\infty}\) and \((H_n)_{n=1}^{\infty}\) such that \noindent (1) for each \(n\in \mathbb{N}\), \(m(n)\in \mathbb{N}\), \(a\in S^{m(n)+1}\), \(H_n\in \mathcal{J}_{m(n)}\), \((\bigcup\limits_{i=1}^{m(n)}H_{i,m(n)})\bigcap(\bigcup\limits_{i=1}^{m(n)+1}H_{n+1,i})=\emptyset\), and \noindent (2) for each \(u\in\Phi\), \(FP((\prod\limits_{j=1}^{m(n)}(a_n(j)\cdot\prod\limits_{t\in H_n(j)}f_{u(n)}(t))\cdot a_n(m+1))_{n=1}^{\infty})\subseteq A\). (Here \(FP\) denotes finite products.)