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Semigroup Forum

, Volume 83, Issue 2, pp 289–303 | Cite as

On the separator of subsets of semigroups

  • Attila NagyEmail author
RESEARCH ARTICLE

Abstract

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 xAA, AxA, x(SA)⊆(SA), (SA)x⊆(SA). 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 IS 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 ∅≠AS)] 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 ∅⊂AS 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.

Keywords

Semigroup Separator of a subset of a semigroup Permutative semigroups 

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References

  1. 1.
    Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. I. AMS, Providence (1961) zbMATHGoogle Scholar
  2. 2.
    Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. II. AMS, Providence (1967) zbMATHGoogle Scholar
  3. 3.
    Nagy, A.: The separator of a subset of a semigroup. Publ. Math. (Debr.) 27(1–2), 25–30 (1980) zbMATHGoogle Scholar
  4. 4.
    Nagy, A.: Special Classes of Semigroups. Kluwer Academic, Dordrecht (2001) zbMATHGoogle Scholar
  5. 5.
    Nagy, A.: On commutative monoid congruences of semigroups. Pure Math. Appl. 13(3), 389–392 (2002) MathSciNetGoogle Scholar
  6. 6.
    Nagy, A., Jones, P.R.: Permutative semigroups whose congruences form a chain. Semigroup Forum 69, 446–456 (2004) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Nordahl, T.E.: On permutative semigroup algebras. Algebra Univers. 25, 322–333 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Putcha, M.S.: Semilattice decomposition of semigroups. Semigroup Forum 6, 12–34 (1973) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Putcha, M.S., Yaqub, A.: Semigroups satisfying permutation identities. Semigroup Forum 3, 68–73 (1971) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Restivo, A., Reutenauer, C.: On the Burnside problem for semigroups. J. Algebra 89, 102–104 (1984) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Reutenauer, C.: Semisimplicity of the algebra associated to a biprefix code. Semigroup Forum 23, 327–342 (1981) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Shevrin, L.N.: Completely simple semigroups without zero and idealizators of subsemigroups. Izv. Vysš. Učebn. Zaved., Mat. 55, 157–160 (1966) (Russian) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Algebra, Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary

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