{"title":"Finite semigroups and periodic sums systems in $$\\beta \\mathbb {N}$$ and their Ramsey theoretic consequences","authors":"Yevhen Zelenyuk","doi":"10.1007/s00233-024-10424-y","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(m,n\\ge 2\\)</span> and define <span>\\(\\nu :\\omega \\rightarrow \\{0,\\ldots ,m-1\\}\\)</span> by <span>\\(\\nu (k)\\equiv k\\pmod {m}\\)</span>. We construct some new finite semigroups in <span>\\(\\beta \\mathbb {N}\\)</span>, in particular, a semigroup generated by <i>m</i> elements of order <i>n</i> with cardinality <span>\\(m^n+m^{n-1}+\\cdots +m\\)</span>. We also show that, for <span>\\(n\\ge m\\)</span>, there is a sequence <span>\\(p_0,\\ldots ,p_{m-1}\\)</span> in <span>\\(\\beta \\mathbb {N}\\)</span> such that all sums <span>\\(\\sum _{j=i}^{i+k}p_{\\nu (j)}\\)</span>, where <span>\\(i\\in \\{0,\\ldots ,m-1\\}\\)</span> and <span>\\(k\\in \\{0,\\ldots ,n-1\\}\\)</span>, are distinct and <span>\\(\\sum _{j=i}^{i+n}p_{\\nu (j)}=\\sum _{j=i}^{i+n-m}p_{\\nu (j)}\\)</span> for each <i>i</i>. As consequences we derive some new Ramsey theoretic results. In particular, we show that, for <span>\\(n\\ge m\\)</span>, there is a partition <span>\\(\\{A_{i,k}:(i,k)\\in \\{0,\\ldots ,m-1\\}\\times \\{0,\\ldots ,n-1\\}\\}\\)</span> of <span>\\(\\mathbb {N}\\)</span> such that, whenever for each (<i>i</i>, <i>k</i>), <span>\\(\\mathscr {B}_{i,k}\\)</span> is a finite partition of <span>\\(A_{i,k}\\)</span>, there exist <span>\\(B_{i,k}\\in \\mathscr {B}_{i,k}\\)</span> and a sequence <span>\\((x_j)_{j=0}^\\infty \\)</span> such that for every finite sequence <span>\\(j_0<\\ldots <j_s\\)</span> such that <span>\\(j_{t+1}\\equiv j_t+1\\pmod {m}\\)</span> for each <span>\\(t<s\\)</span>, one has <span>\\(x_{j_0}+\\cdots +x_{j_s}\\in B_{i_0,k_0}\\)</span>, where <span>\\(i_0=\\nu (j_0)\\)</span> and <span>\\(k_0\\)</span> is <i>s</i> if <span>\\(s\\le n-1\\)</span> and <span>\\(n-m+\\nu (s-n)\\)</span> otherwise.</p>","PeriodicalId":49549,"journal":{"name":"Semigroup Forum","volume":"75 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Semigroup Forum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00233-024-10424-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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.
让(m,nge 2)定义(\nu :\omega \rightarrow \{0,\ldots,m-1})为(\nu (k)\equiv k\pmod {m})。我们在 \(\beta \mathbb {N}/)中构造了一些新的有限半群,特别是由阶数为 n 的 m 个元素产生的半群,它的心数为(m^n+m^{n-1}+\cdots +m\ )。我们还证明,对于(nge m),在(beta \mathbb {N})中有一个序列(p_0,\ldots ,p_{m-1}\),使得所有的和(sum _{j=i}^{i+k}p_{\nu (j)}\)、其中(i在{0,\ldots ,m-1\}\中)和(k在{0,\ldots ,n-1\}\中)是不同的,并且(对于每个i来说,(sum _{j=i}^{i+n}p_{\nu (j)}=sum _{j=i}^{i+n-m}p_{\nu (j)}\ )是不同的。因此,我们得出了一些新的拉姆齐理论结果。特别是,我们证明了,对于(n\ge m\ ),存在一个分割(\{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 \(j_0<;\dots <j_s\) such that \(j_{t+1}\equiv j_t+1\pmod {m}\) for each \(t<;s),就有\(x_{j_0}+\cdots +x_{j_s}\in B_{i_0,k_0}\),其中\(i_0=\nu (j_0)\) and \(k_0\) is s if \(s\le n-1\) and\(n-m+\nu (s-n)\) otherwise.
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