Semigroup Forum

, Volume 6, Issue 1, pp 59–68 | Cite as

Categorical semigroups

  • R. A. R. Monzo


This investigation was stimulated by a question raised by F.R. McMorris and M. Satyanarayana [Proc. Amer. Math. Soc.33 (1972), 271–277] which asked whether a regular semigroup with a tree of idempotents is categorical. The question is answered in the affirmative. Characterizations of categorical semigroups are found within the following classes of semigroups: regular semigroups, bands, commutative regular semigroups, unions of simple semigroups, semilattices of groups, and commutative semigroups.

Some results are related to part of the work of M. Petrich [Trans. Amer. Math. Soc.170 (1972), 245–268]. For instance, it is shown that the poset of J-classes of any regular categorical semigroup is a tree; however, an example of a regular non-categorical semigroup is given in which the poset of J-classes is a chain.

It is also shown that the condition that the subsemigroup of idempotents be categorical is sufficient, but not necessary, for an orthodox semigroup to be categorical.


Inverse Semigroup Regular Semigroup Homomorphic Image Commutative Semigroup Simple Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Clifford, A.H. and G.B. Preston,The Algebraic Theory of Semigroups, Volumes I and II (Math. Surveys7 (I and II). Amer. Math. Soc., Providence, Rhode Island, 1961 and 1967).MATHGoogle Scholar
  2. [2]
    Hall, T.E.,On orthodox semigroups and uniform and antiuniform bands, J. Algebra16 (1970), 204–217.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    McMorris, F.R. and M. Satyanarayana,Categorical semigroups, Proc. Amer. Math. Soc. (1972), 271–277.Google Scholar
  4. [4]
    Munn, W.D.,Uniform semilattices and bisimple inverse semigroups, Quart. J. Math. Oxford (2)17 (1966), 151–159.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Petrich, M.,Regular semigroups satisfying certain conditions on idempotents and ideals, Trans. Amer. Math. Soc.170 (1972), 245–268.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Petrich, M.,The maximal semilattice decomposition of a semigroup, Math. Z.85 (1964), 68–82.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Tamura, T.,Semigroups satisfying identity xy=f(x, y), Pacific J. Math.31 (1962), 513–521.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1973

Authors and Affiliations

  • R. A. R. Monzo
    • 1
  1. 1.Department of Mathematics, Institute of Advanced StudiesAustralian National UniversityCanberraAustralia

Personalised recommendations