J. Aust. Math. Soc. 83 (2007), no. 1, pp. 87–104.
Canonical varieties of completely regular semigroups
Mario Petrich
21420 Bol, Brač
Received 15 April 2002; revised 1 August 2005
Communicated by D. Easdown


Completely regular semigroups \mathcal {CR} are regarded here as algebras with multiplication and the unary operation of inversion. Their lattice of varieties is denoted by \mathcal L (\mathcal {CR}). Let \mathcal B denote the variety of bands and \mathcal L (\mathcal B) the lattice of its subvarieties. The mapping \mathcal V \rightarrow \mathcal V \cap \mathcal B is a complete homomorphism of \mathcal L (\mathcal {CR}) onto \mathcal L(\mathcal B). The congruence induced by it has classes that are intervals, say \mathcal {V}B=[\mathcal V_B,\mathcal V^B] for \mathcal V \in \mathcal L(\mathcal {CR}). Here \mathcal V_B=\mathcal V \cap \mathcal B. We characterize \mathcal V^B in several ways, the principal one being an inductive way of constructing bases for \vee -irreducible band varieties. We term the latter canonical. We perform a similar analysis for the intersection of these varieties with the varieties \mathcal {BG}, \mathcal {OBG} and \mathcal B.

Download the article in PDF format (size 204 Kb)

2000 Mathematics Subject Classification: primary 20M07
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2354??? Z'blatt-MATH: pre05231334


  1. J. A. Gerhard and M. Petrich, ‘Certain characterizations of varieties of bands’, Proc. Edinburgh Math. Soc. (2) 31 (1988), 301–319. MR989763
  2. J. A. Gerhard and M. Petrich, ‘Varieties of bands revisited’, Proc. London Math. Soc. (3) 58 (1989), 323–350. MR977480
  3. F. Pastijn and M. Petrich, ‘Congruences on regular semigroups’, Trans. Amer. Math. Soc. 295 (1986), 607–633. MR833699
  4. F. J. Pastijn, ‘The lattice of completely regular semigroup varieties’, J. Aust. Math. Soc. 49 (1990), 24–42. MR1054080
  5. M. Petrich, ‘Varieties of orthodox bands of groups’, Pacific J. Math. 58 (1975), 209–217. MR382522
  6. M. Petrich and N. R. Reilly, ‘Operators related to E-disjunctive and fundamental completely regular semigroups’, J. Algebra 134 (1990), 1–27. MR1068411
  7. M. Petrich and N. R. Reilly, Completely regular semigroups, Vol. I, Canadian Math. Soc. Series of Monographs and Advanced Texts 23 (Wiley, New York, 1999). MR1684919
  8. M. Petrich and N. R. Reilly, Completely regular semigroups, Vol. II (in preparation).
  9. L. Polák, ‘On varieties of completely regular semigroups I’, Semigroup Forum 32 (1985), 97–123. MR803483
  10. L. Polák, ‘On varieties of completely regular semigroups II’, Semigroup Forum 36 (1987), 253–284. MR916425
  11. V. V. Rasin, ‘Varieties of orthodox Clifford semigroups’, Izv. Vyssh. Uchebn. Zaved. Matem. 11 (1982), 82–85 (Russian). MR687319
  12. N. R. Reilly, ‘Completely regular semigroups’, in: Lattices, Semigroups and Universal Algebra (ed. J. Almeida) (Plenum Press, New York, 1990) pp. 225–242. MR1085084
  13. N. R. Reilly and S. Zhang, ‘Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands’, Algebra Universalis 44 (2000), 217–239. MR1816020
  14. P. G. Trotter, ‘Subdirect decomposition of the lattice of varieties of completely regular semigroups’, Bull. Austral. Math. Soc. 39 (1989), 343–351. MR995132
  15. P. G. Trotter and P. Weil, ‘The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue’, Algebra Universalis 37 (1997), 491–526. MR1465305