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- Title
Finite semigroups and periodic sums systems in βN and their Ramsey theoretic consequences.
- Authors
Zelenyuk, Yevhen
- Abstract
Let m , n ≥ 2 and define ν : ω → { 0 , ... , m - 1 } by ν (k) ≡ k (mod m) . We construct some new finite semigroups in β N , in particular, a semigroup generated by m elements of order n with cardinality m n + m n - 1 + ⋯ + m . We also show that, for n ≥ m , there is a sequence p 0 , ... , p m - 1 in β N such that all sums ∑ j = i i + k p ν (j) , where i ∈ { 0 , ... , m - 1 } and k ∈ { 0 , ... , n - 1 } , are distinct and ∑ j = i i + n p ν (j) = ∑ j = i i + n - m p ν (j) for each i. As consequences we derive some new Ramsey theoretic results. In particular, we show that, for n ≥ m , there is a partition { A i , k : (i , k) ∈ { 0 , ... , m - 1 } × { 0 , ... , n - 1 } } of N such that, whenever for each (i, k), B i , k is a finite partition of A i , k , there exist B i , k ∈ B i , k and a sequence (x j) j = 0 ∞ such that for every finite sequence j 0 < ... < j s such that j t + 1 ≡ j t + 1 (mod m) for each t < s , one has x j 0 + ⋯ + x j s ∈ B i 0 , k 0 , where i 0 = ν (j 0) and k 0 is s if s ≤ n - 1 and n - m + ν (s - n) otherwise.
- Subjects
RAMSEY theory; RAMSEY numbers; SEQUENCE spaces
- Publication
Semigroup Forum, 2024, Vol 108, Issue 2, p488
- ISSN
0037-1912
- Publication type
Article
- DOI
10.1007/s00233-024-10424-y