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

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