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Further results in the theory of generalised inflations of semigroups

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Abstract

We determine the structure of semigroups that satisfy xyzw∈{xy,xw,zy,zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F(X) on a set X. In the latter case the semigroup must be an inflation of F(X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band.

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References

  1. Clarke, G.T., Monzo, R.A.R.: A generalisation of the concept of an inflation of a semigroup. Semigroup Forum 60, 172–186 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Math. Surveys of the American Math. Soc., vol. 1, pp. 126–128. Am. Math. Soc., Providence (1961)

    MATH  Google Scholar 

  3. Giraldes, E., Howie, J.M.: Semigroups of high rank. Proc. Edinb. Math. Soc. 28, 13–34 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hall, T.E.: Almost commutative bands. Glasg. Math. J. 13, 176–178 (1972)

    MATH  Google Scholar 

  5. Howie, J.M.: An Introduction to Semigroup Theory, p. 98. Academic Press, San Diego (1976)

    MATH  Google Scholar 

  6. McLean, D.: Idempotent semigroups. Am. Math. Mon. 61, 110–113 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  7. McMorris, F.R., Satyanarayana, M.: Categorical semigroups. Proc. Am. Math. Soc. 271–277 (1972)

  8. Monzo, R.A.R.: Categorical semigroups. Semigroup Forum 6, 59–68 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. Monzo, R.A.R.: Pre-compatible almost endomorphisms and semigroups whose cube is a band. Semigroup Forum 67, 355–372 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pelikan, J.: On semigroups in which products are equal to one of the factors. Period. Math. Hung. 4, 103–106 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Petrich, M.: A simple construction of semigroups all of whose subsemigroups are left ideals. Semigroup Forum 4, 262–266 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pondelicek, B.: On Semigroups Having Regular Globals. Colloq. Mathem. Soc. Janos Bolyai (1976)

  13. Redei, L.: Algebra I. Pergamon, Elmsford (1967)

    Google Scholar 

  14. Tamura, T.: Semigroups satisfying identity xy=f(x,y). Pac. J. Math. 31, 513–521 (1962)

    Google Scholar 

  15. Wang, Q., Wismath, S.L.: Null extensions and generalised inflations of Brandt semigroups. Semigroup Forum 74, 274–292 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yamada, M.: A note on middle unitary semigroups. Kodai Math. Semin. Rep. 7, 49–52 (1955)

    Article  Google Scholar 

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  1. 10 Albert Mansions, Crouch Hill, London, N8 9RE, UK

    R. A. R. Monzo

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  1. R. A. R. Monzo
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Correspondence to R. A. R. Monzo.

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Communicated by Thomas E. Hall.

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Monzo, R.A.R. Further results in the theory of generalised inflations of semigroups. Semigroup Forum 76, 540–560 (2008). https://doi.org/10.1007/s00233-008-9055-7

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  • DOI: https://doi.org/10.1007/s00233-008-9055-7

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