A note-question on partitions of semigroups

I. Protasov; K. Protasova

doi:10.15330/ms.44.1.104-106

Given a semigroup $S$ and an $n$-partition $\mathcal{P}$ of $S$, $n\in \mathbb{N}$, do there exist $A\in \mathcal{P}$ and a subset~$F$ of $S$ such that $S=F ^{-1} \{x \in S\colon x A \cap A\neq\emptyset\}$ and $|F |\leq n$? We give an affirmative answer provided that either $S$ is finite or $n=2$.

1. Banakh T., Protasov I., Slobodianiuk S. Densities, submeasures and partitions of G-spaces and groups, Algebra and Discrete Mathematics, 17 (2014), №2, 193–221; available at http://arxiv.org./abs/1303.4612.

2. Hindman N., Strauss D. Algebra in the Stone-Cech Compactification. – 2nd edition, de Gruyter, 2012.

3. Mazurov V.D., Khukhro E.I. (eds), Unsolved problems in group theory, the Kourovka notebook. – 13-th augmented edition, Novosibirsk, 1995.

4. Protasov I., Banakh T. Ball structures and colorings of groups and graphs. – Math. Stud. Monogr. Ser., V.11, VNTL, Lviv, 2003.

	A note-question on partitions of semigroups
Author	I. Protasov, K. Protasova I.V.Protasov@gmail.com Department of Cybernetics, Kyiv University
Abstract	Given a semigroup $S$ and an $n$-partition $\mathcal{P}$ of $S$, $n\in \mathbb{N}$, do there exist $A\in \mathcal{P}$ and a subset~$F$ of $S$ such that $S=F ^{-1} \{x \in S\colon x A \cap A\neq\emptyset\}$ and $\|F \|\leq n$? We give an affirmative answer provided that either $S$ is finite or $n=2$.
Keywords	partitions of semigroups; covering number
DOI	doi:10.15330/ms.44.1.104-106
Reference	1. Banakh T., Protasov I., Slobodianiuk S. Densities, submeasures and partitions of G-spaces and groups, Algebra and Discrete Mathematics, 17 (2014), №2, 193–221; available at http://arxiv.org./abs/1303.4612. 2. Hindman N., Strauss D. Algebra in the Stone-Cech Compactification. – 2nd edition, de Gruyter, 2012. 3. Mazurov V.D., Khukhro E.I. (eds), Unsolved problems in group theory, the Kourovka notebook. – 13-th augmented edition, Novosibirsk, 1995. 4. Protasov I., Banakh T. Ball structures and colorings of groups and graphs. – Math. Stud. Monogr. Ser., V.11, VNTL, Lviv, 2003.
Pages	104-106
Volume	44
Issue	1
Year	2015
Journal	Matematychni Studii
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