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

IF 0.7 3区 数学 Q2 MATHEMATICS
Yevhen Zelenyuk
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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 (ik), \(\mathscr {B}_{i,k}\) 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.

$$\beta \mathbb {N}$ 中的有限半群和周期和系统及其拉姆齐理论后果
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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来源期刊
Semigroup Forum
Semigroup Forum 数学-数学
CiteScore
1.50
自引率
14.30%
发文量
79
审稿时长
12 months
期刊介绍: Semigroup Forum is a platform for speedy and efficient transmission of information on current research in semigroup theory. Scope: Algebraic semigroups, topological semigroups, partially ordered semigroups, semigroups of measures and harmonic analysis on semigroups, numerical semigroups, transformation semigroups, semigroups of operators, and applications of semigroup theory to other disciplines such as ring theory, category theory, automata, logic, etc. Languages: English (preferred), French, German, Russian. Survey Articles: Expository, such as a symposium lecture. Of any length. May include original work, but should present the nonspecialist with a reasonably elementary and self-contained account of the fundamental parts of the subject. Research Articles: Will be subject to the usual refereeing procedure. Research Announcements: Description, limited to eight pages, of new results, mostly without proofs, of full length papers appearing elsewhere. The announcement must be accompanied by a copy of the unabridged version. Short Notes: (Maximum 4 pages) Worthy of the readers'' attention, such as new proofs, significant generalizations of known facts, comments on unsolved problems, historical remarks, etc. Research Problems: Unsolved research problems. Announcements: Of conferences, seminars, and symposia on Semigroup Theory. Abstracts and Bibliographical Items: Abstracts in English, limited to one page, of completed work are solicited. Listings of books, papers, and lecture notes previously published elsewhere and, above all, of new papers for which preprints are available are solicited from all authors. Abstracts for Reviewing Journals: Authors are invited to provide with their manuscript informally a one-page abstract of their contribution with key words and phrases and with subject matter classification. This material will be forwarded to Zentralblatt für Mathematik.
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