Characterization of elements of polynomials in \(\beta S\)
In this paper, the author characterizes the members of polynomials evaluated at idempotents in the Stone-Čech compactification \(\beta{\mathbb N}\) of the discrete semigroup of natural numbers \(\mathbb N\). In fact, he obtains the following more general results. The set of all nonempty finite subsets of a set \(X\) is denoted by \(P_f(X)\). Let \((S, +)\) and \((S,\cdot)\) be discrete semigroups. For each sequence \(\langle x_t\rangle_{t=1}^{\infty}\) in \(S\) and \(m\in{\mathbb N}\), let \(FS(\langle x_t\rangle_{t=m}^{\infty})= \big\{\sum_{t\in F}x_t:F\in P_f(\{x_i:i\geq m\})\big\}\). For a polynomial \(g(z_1,z_2,\dots,z_k)\) with coefficients from \(S\) and \(m\) the number of occurrences of variables in \(g\), let \(h_g(y_1,y_2,\dots,y_m)\) the polynomial obtained from \(g\) by replacing the \(i^{\,\text{th}}\) occurrence of a variable by \(y_i\), and define the function \(f_g:\{1,2,\dots,m\}\to\{1,2,\dots,k\}\) by \(f_g(i)=j\) if the \(i^{\,\text{th}}\) occurrence of a variable is \(z_j\). Applying the notation, one has that \[ g(z_1,z_2,\dots,z_k)=h_g(z_{f_g(1)},z_{f_g(2)},\dots,z_{f_g(m)}). \] For a polynomial \(g(z_1,z_2,\dots,z_k)\) with coefficients from \(S\) and no constant term, the following statements hold: (1) For \(j\in\{1,2,\dots,k\}\), let \(\langle x_{j,t}\rangle_{t=1}^{\infty}\) be a sequence in \(S\) and \(p_i\in\bigcap_{l=1}^{\infty}\overline{FS(\langle x_{j,t}\rangle _{t=l}^{\infty})}^{\beta S}\). Then \[ \Big\{h_g\Big(\sum_{t\in F_1}x_{f_g(1),t}, \dots,\sum_{t\in F_m}x_{f_g(m),t}\Big): \max F_i<\min F_{i+1}, \;F_i\in P_f({\mathbb N})\Big\} \in g(p_1,p_2,\dots,p_k). \] (2) For \(i\in\{1,2,\dots,k\}\), let \(\langle x_{i,t}\rangle_{t=1}^{\infty}\) be a sequence in \(S\) and \(p_i\) an idempotent in \((\beta S,+)\) such that \(p_i\in\bigcap_{m=1}^{\infty} \overline{FS(\langle x_{i,n}\rangle_{n=m}^{\infty})}^{\beta S}\) and \(A\in g(p_1,p_2,\dots,p_k)\). Then for each \(i=1,2,\dots,k\), there exist sequences \(\langle y_{i,n}\rangle_{n=1}^{\infty}\) in \(S\) and \(\langle H_t\rangle_{t=1}^{\infty}\) in \(P_f({\mathbb N})\) such that for each \(t\in{\mathbb N}\) \(\max H_t<\min H_{t+1}\) and \(y_t=\sum_{n\in H_t}x_n\), and \[ \Big\{h_g\Big(\sum_{t\in F_1}y_{f_g(1),t}, \dots,\sum_{t\in F_m}y_{f_g(m),t}\Big): \max F_i<\min F_{i+1}, \;F_i\in P_f({\mathbb N})\Big\}\in A. \]
scientific article; zbMATH DE number 6018051