Czechoslovak Mathematical Journal

, Volume 63, Issue 2, pp 289–305 | Cite as

Normal cryptogroups with an associate subgroup

  • Mario Petrich


Let S be a semigroup. For a, xS such that a = axa, we say that x is an associate of a. A subgroup G of S which contains exactly one associate of each element of S is called an associate subgroup of S. It induces a unary operation in an obvious way, and we speak of a unary semigroup satisfying three simple axioms.

A normal cryptogroup S is a completely regular semigroup whose H -relation is a congruence and S/H is a normal band. Using the representation of S as a strong semilattice of Rees matrix semigroups, in a previous communication we characterized those that have an associate subgroup.

In this paper, we use that result to find three more representations of this semigroup. The main one has a form akin to the one of semigroups in which the identity element of the associate subgroup is medial.


semigroup normal cryptogroup associate subgroup representation strong semilattice of semigroups Rees matrix semigroup 

MSC 2010



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  1. [1]
    T. S. Blyth, P. M. Martins: On associate subgroups of regular semigroups. Commun. Algebra 25 (1997), 2147–2156.MATHCrossRefGoogle Scholar
  2. [2]
    P. M. Martins, M. Petrich: Unary semigroups with an associate subgroup. Commun. Algebra 36 (2008), 1999–2013.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    M. Petrich: The existence of an associate subgroup in normal cryptogroups. Publ. Math. Debrecen 73 (2008), 281–298.MathSciNetMATHGoogle Scholar
  4. [4]
    M. Petrich, N. R. Reilly: Completely Regular Semigroups. Canadian Mathematical Society Series of Monographs and Advanced Texts 23, John Wiley & Sons, New York, 1999.MATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2013

Authors and Affiliations

  • Mario Petrich
    • 1
  1. 1.Bol, BračCroatia

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