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Semigroup Forum

, Volume 91, Issue 1, pp 295–298 | Cite as

A natural characterization of semilattices of rectangular bands and groups of exponent two

SHORT NOTE

Abstract

In a recent paper, Monzo characterized semilattices of rectangular bands and groups of exponent \(2\) as the semigroups that satisfy the following conditions: \(x = x^{3}\) and \(xyx \in \{xy^{2}x, y^{2}xy^{2}\}\). However, this definition does not seem to point directly to the properties of rectangular bands and groups of exponent \(2\) (namely, idempotency and commutativity). So, in order to provide a more natural characterization of the class of semigroups under consideration we prove the following theorem: Main Theorem In a semigroup \(S\), the following are equivalent:
  • \(S\) is a semilattice of rectangular bands and groups of exponent \(2\);

  • for all \(x,y \in S\), we have \(x = x^{3} and xy \in \{yx, (xy)^{2}\}\).

The paper ends with a list of problems.

Keywords

Rectangular bands Groups of exponent two 

Notes

Acknowledgments

The second author found Theorem 2.2 with the assistance of the automated deduction tool Prover9 developed by McCune [4]. Then the first author found a human proof for the result.

References

  1. 1.
    Howie, John M.: Fundamentals of Semigroup Theory. Oxford Science, Oxford (1995)MATHGoogle Scholar
  2. 2.
    Petrich, M., Reilly, N.R.: Completely Regular Semigroups. Wiley-Blackwell, New York (1999)MATHGoogle Scholar
  3. 3.
    Monzo, R.A.R.: Semilattices of Rectangular Bands and Groups of Order 2. (2010) http://arxiv.org/ftp/arxiv/papers/1301/1301.0828.Accessed 6 Nov 2013
  4. 4.
    McCune, W.: Prover9 and Mace4, version 2009–11A, www.cs.unm.edu/mccune/prover9/. Accessed 6 Nov 2013

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Colégio PlanaltoLisboaPortugal
  2. 2.Department of MathematicsUniversity of DenverDenverUSA

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