Finite semigroups and periodic sums systems in \(\beta \mathbb {N}\) and their Ramsey theoretic consequences

Let \(m,n\ge 2\) $m,n\ge 2$ and define \(\nu :\omega \rightarrow \{0,\ldots ,m-1\}\) $\nu :\omega \to \{0,\dots ,m-1\}$ by \(\nu (k)\equiv k\pmod {m}\) $\nu (k)\equiv k(modm)$ . We construct some new finite semigroups in \(\beta \mathbb {N}\) $\beta \mathbb{N}$ , in particular, a semigroup generated by m elements of order n with cardinality \(m^n+m^{n-1}+\cdots +m\) $m^n+m^{n-1}+\cdots +m$ . We also show that, for \(n\ge m\) $n\ge m$ , there is a sequence \(p_0,\ldots ,p_{m-1}\) $p_0,\dots ,p_{m-1}$ in \(\beta \mathbb {N}\) $\beta \mathbb{N}$ such that all sums \(\sum _{j=i}^{i+k}p_{\nu (j)}\) ${\sum }_{j=i}^{i+k}{p}_{\nu (j)}$ , where \(i\in \{0,\ldots ,m-1\}\) $i\in \{0,\dots ,m-1\}$ and \(k\in \{0,\ldots ,n-1\}\) $k\in \{0,\dots ,n-1\}$ , are distinct and \(\sum _{j=i}^{i+n}p_{\nu (j)}=\sum _{j=i}^{i+n-m}p_{\nu (j)}\) ${\sum }_{j=i}^{i+n}{p}_{\nu (j)}={\sum }_{j=i}^{i+n-m}{p}_{\nu (j)}$ for each i. As consequences we derive some new Ramsey theoretic results. In particular, we show that, for \(n\ge m\) $n\ge m$ , there is a partition \(\{A_{i,k}:(i,k)\in \{0,\ldots ,m-1\}\times \{0,\ldots ,n-1\}\}\) $\{A_{i,k}:(i,k)\in \{0,\dots ,m-1\}\times \{0,\dots ,n-1\}\}$ of \(\mathbb {N}\) $\mathbb{N}$ such that, whenever for each (i, k), \(\mathscr {B}_{i,k}\) ${\mathcal{B}}_{i,k}$ is a finite partition of \(A_{i,k}\) $A_{i,k}$ , there exist \(B_{i,k}\in \mathscr {B}_{i,k}\) $B_{i,k}\in {\mathcal{B}}_{i,k}$ and a sequence \((x_j)_{j=0}^\infty \) ${(x_j)}_{j=0}^{\infty }$ such that for every finite sequence \(j_0<\ldots <j_s\) $j_0<\dots <j_s$ such that \(j_{t+1}\equiv j_t+1\pmod {m}\) $j_{t+1}\equiv j_t+1(modm)$ for each \(t<s\) $t<s$ , one has \(x_{j_0}+\cdots +x_{j_s}\in B_{i_0,k_0}\) $x_{j_0}+\cdots +x_{j_s}\in B_{i_0,k_0}$ , where \(i_0=\nu (j_0)\) $i_0=\nu (j_0)$ and \(k_0\) $k_0$ is s if \(s\le n-1\) $s\le n-1$ and \(n-m+\nu (s-n)\) $n-m+\nu (s-n)$ otherwise.