caa a22 4500 44534542X CHVBK 20180317142827.0 cr unu---uuuuu 170323e20111001xx s 000 0 eng 10.1007/s00233-011-9306-x doi (NATIONALLICENCE)springer-10.1007/s00233-011-9306-x Nagy Attila Department of Algebra, Institute of Mathematics, Budapest University of Technology and Economics, Pf. 91, 1521, Budapest, Hungary aut On the separator of subsets of semigroups [Elektronische Daten] [Attila Nagy] By the separator $\operatorname{\mathit{Sep}}A$ of a subset A of a semigroup S we mean the set of all elements x of S which satisfy conditions xA⊆A, Ax⊆A, x(S−A)⊆(S−A), (S−A)x⊆(S−A). In this paper we deal with the separator of subsets of semigroups. In Sect.2, we investigate the separator of subsets of special types of semigroups. We prove that, in the multiplicative semigroup S of all n×n matrices over a field ${\mathbb{F}}$ and in the semigroup S of all transformations of a set X with |X|=n<∞, the separator of an ideal I≠S is the unit group of S. We show that the separator of a subset of a group G is a subgroup of G. Moreover, the separator of a subset A of a completely regular semigroup [a Clifford semigroup; a completely 0-simple semigroup] S is either empty or a completely regular semigroup [a Clifford semigroup, a completely simple semigroup (supposing ∅≠A≠S)] of S. In Sect.3 we characterize semigroups which satisfy certain conditions concerning the separator of their subsets. We prove that every subset A of a semigroup S with ∅⊂A⊂S has an empty separator if and only if S is an ideal extension of a rectangular band by a nil semigroup. We also prove that every subsemigroup of a semigroup S is the separator of some subset of S if and only if $\emptyset \subset \operatorname{\mathit{Sep}}A\subseteq A$ is satisfied for every subsemigroup A of S if and only if S is a periodic group if and only if $A=\operatorname{\mathit{Sep}}A$ for every subsemigroup A of S. In Sect.4 we apply the results of Sect.3 for permutative semigroups. We show that every permutative semigroup without idempotent elements has a non-trivial group or a group with a zero homomorphic image. Moreover, if a finitely generated permutative semigroup S has no neither a non-trivial group homomorphic image nor a group with a zero homomorphic image then S is finite. Springer Science+Business Media, LLC, 2011 Semigroup nationallicence Separator of a subset of a semigroup nationallicence Permutative semigroups nationallicence Semigroup Forum Springer-Verlag 83/2(2011-10-01), 289-303 0037-1912 83:2<289 2011 83 233 https://doi.org/10.1007/s00233-011-9306-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00233-011-9306-x text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Nagy Attila Department of Algebra, Institute of Mathematics, Budapest University of Technology and Economics, Pf. 91, 1521, Budapest, Hungary aut NATIONALLICENCE 773 0- Semigroup Forum Springer-Verlag 83/2(2011-10-01), 289-303 0037-1912 83:2<289 2011 83 233 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer