Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Semidirect products of regular semigroups


Authors: Peter R. Jones and Peter G. Trotter
Journal: Trans. Amer. Math. Soc. 349 (1997), 4265-4310
MSC (1991): Primary 20M17, 20M07
DOI: https://doi.org/10.1090/S0002-9947-97-01638-3
MathSciNet review: 1355299
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Within the usual semidirect product $S*T$ of regular semigroups $S$ and $T$ lies the set $\text {Reg}\,% (S*T)$ of its regular elements. Whenever $S$ or $T$ is completely simple, $\text {Reg}\,% (S*T)$ is a (regular) subsemigroup. It is this `product' that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, ${\mathbf U}% $ and ${\mathbf V}% $, the e-variety ${\mathbf U}*{\mathbf V}% $ generated by $\{\text {Reg}\,% (S*T) : S \in {\mathbf U}% , T \in {\mathbf V}% \}$ is well defined if and only if either ${\mathbf U} % $ or ${\mathbf V}% $ is contained within the e-variety ${\mathbf {CS}}% $ of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety $L{\mathbf I}% $ of locally inverse semigroups is decomposed as ${\mathbf I}% * {\mathbf {RZ}}% $, where ${\mathbf I}% $ is the variety of inverse semigroups and ${\mathbf {RZ}}% $ is that of right zero semigroups; and the e-variety ${\mathbf {ES}}% $ of $E$-solid semigroups is decomposed as ${\mathbf {CR}}*{\mathbf G}% $, where ${\mathbf {CR}}% $ is the variety of completely regular semigroups and ${\mathbf G}% $ is the variety of groups. In the second half of the paper, a general construction is given for the e-free semigroups (the analogues of free semigroups in this context) in a wide class of semidirect products ${\mathbf U}% * {\mathbf V}% $ of the above type, as a semidirect product of e-free semigroups from ${\mathbf U}% $ and ${\mathbf V}% $, ``cut down to regular generators''. Included as special cases are the e-free semigroups in almost all the known important e-varieties, together with a host of new instances. For example, the e-free locally inverse semigroups, $E$-solid semigroups, orthodox semigroups and inverse semigroups are included, as are the e-free semigroups in such sub-e-varieties as strict regular semigroups, $E$-solid semigroups for which the subgroups of its self-conjugate core lie in some given group variety, and certain important varieties of completely regular semigroups. Graphical techniques play an important role, both in obtaining decompositions and in refining the descriptions of the e-free semigroups in some e-varieties. Similar techniques are also applied to describe the e-free semigroups in a different `semidirect' product of e-varieties, recently introduced by Auinger and Polák. The two products are then compared.


References [Enhancements On Off] (What's this?)

  • 1. J. Almeida, Semidirect products of semigroups from the universal algebraist's point of view, J. Pure Appl. Algebra 60 (1989), 113-128. MR 91a:20068
  • 2. K. Auinger, The word problem for the bifree combinatorial strict regular semigroup, Math. Proc. Cambridge Phil. Soc. 113 (1993), 519-533. MR 94b:20061
  • 3. K. Auinger, The bifree locally inverse semigroup on a set, J. Algebra 166 (1994), 630-650. MR 95h:20081
  • 4. K. Auinger, On the bifree locally inverse semigroup, J. Algebra 178 (1995), 581-613. MR 96i:20073
  • 5. K. Auinger and L. Polák, A multiplication of existence varieties of locally inverse semigroups, preprint.
  • 6. S. Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, New York, 1976. MR 58:26604b
  • 7. T.E. Hall, Some properties of local subsemigroups inherited by larger subsemigroups, Semigroup Forum 25 (1982), 35-49. MR 83m:20082
  • 8. T.E. Hall, Identities for existence varieties of regular semigroups, Bull. Austral. Math. Soc. 40 (1989), 59-77. MR 90j:20127
  • 9. T.E. Hall, Regular semigroups: amalgamation and the lattice of existence varieties, Algebra Universalis 29 (1991), 79-108. MR 92g:20098
  • 10. T.E. Hall, A concept of variety for regular semigroups, in `Proceedings of the Monash Conference on Semigroup Theory', T.E. Hall, P.R. Jones J. Meakin eds, World Scientific Publ. Co., Singapore, 1991, 101-116. MR 94g:20083
  • 11. J.M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976. MR 57:6235
  • 12. P.R. Jones, Mal'cev products of varieties of completely regular semigroups, J. Austral. Math. Soc. (Ser. A) 42 (1987), 227-246. MR 88a:20069
  • 13. P.R. Jones, An introduction to existence varieties of regular semigroups, Southeast Asian Bull. Math. 19 (1995), 107-118. MR 96f:20088
  • 14. P.R. Jones, E-free objects and e-locality of completely regular semigroups, Semigroup Forum 51 (1995), 357-377. CMP 96:01
  • 15. P.R. Jones and M.B. Szendrei, Local varieties of completely regular monoids, J. Algebra 150 (1992), 1-27. MR 94g:20084
  • 16. P.R. Jones and P.G. Trotter, Locality of DS and associated varieties, J. Pure Appl. Algebra 104 (1995), 275-301. MR 96i:20074
  • 17. J. Ka\v{d}ourek, On the word problem for free bands of groups and for free objects in some other varieties of completely regular semigroups, Semigroup Forum 38 (1989), 1-55. MR 89k:20085
  • 18. J. Ka\v{d}ourek and L. Polák, On the word problem for free completely regular semigroups, Semigroup Forum 34 (1986), 127-138. MR 88a:20066
  • 19. J. Ka\v{d}ourek and M. Szendrei, A new approach in the theory of orthodox semigroups, Semigroup Forum 40 (1990), 257-296. MR 91k:20075
  • 20. J. Ka\v{d}ourek and M. Szendrei, On existence varieties of $E$-solid semigroups, Semigroup Forum, to appear.
  • 21. S.W. Margolis, Kernels and expansions: a historical and technical perspective, in `Monoids and Semigroups with Applications', J. Rhodes ed., World Scientific Publ. Co., Singapore, 1991, 1-30. MR 93a:20101
  • 22. S.W. Margolis and J.-E. Pin, Inverse semigroups and extensions of semilattices by groups, J. Algebra 110 (1987), 277 - 297. MR 89j:20066a
  • 23. D.B. McAlister, Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups, J. Austral. Math. Soc. Ser. A 31 (1981), 325-336. MR 84d:20062
  • 24. D.B. McAlister, Rees matrix covers for locally inverse semigroups, Trans. Amer. Math. Soc. 277 (1983), 727-738. MR 84m:20073
  • 25. D.B. McAlister, Rees matrix covers for regular semigroups, J. Algebra 89 (1984), 264-279. MR 86b:20077
  • 26. W.R. Nico, On the regularity of semidirect products, J. Algebra 80 (1983), 29-36. MR 84i:20064
  • 27. K.S.S. Nambooripad, The structure of regular semigroups I, Mem. Amer. Math. Soc. 224 (1979). MR 81i:20086
  • 28. H. Neumann, Varieties of Groups, Springer-Verlag, Berlin, 1967. MR 35:6734
  • 29. F. Pastijn, The lattice of completely regular semigroup varieties, J. Austral. Math. Soc. (Ser A) 49 (1990), 24-42. MR 91d:20068
  • 30. F. Pastijn and P.G. Trotter, Lattices of completely regular semigroup varieties, Pacific J. Math. 119 (1985), 191-224. MR 87b:20079
  • 31. M. Petrich, Inverse Semigroups (Wiley, New York, 1984). 59-74. MR 85k:20001
  • 32. L. Polák, A multiplication on the lattice of varieties of *-regular semigroups, in ``Semigroups", C. Bonzini, A. Cherubini and C. Tibiletti, eds, World Sci. Publ. Co., Singapore, 1993, 231-245.
  • 33. M. Szendrei, Orthogroup bivarieties are bilocal, in `Semigroups With Applications', J.M. Howie, W.D. Munn and H.J. Weinert eds, World Scientific Publ. Co., Singapore, 1992, 114-131. MR 93j:20131
  • 34. M. Szendrei, On $E$-unitary covers of orthodox semigroups, Int. J. Alg. Comp. 3 (1993), 317-334. MR 95b:20093
  • 35. M. Szendrei, The bifree regular $E$-solid semigroups, Semigroup Forum 52 (1996), 61-82. MR 96m:20099
  • 36. L.A. Skornyakov, Regularity of the wreath product of monoids, Semigroup Forum 18 (1979), 83-86. MR 80e:20077
  • 37. B. Tilson, Categories as algebra: an essential ingredient in the theory of monoids, J. Pure Appl. Algebra 48 (1987), 83-198. MR 90e:20061
  • 38. P.G. Trotter, Congruence extensions in regular semigroups, J. Algebra 137 (1991), 166-179. MR 92e:20045
  • 39. P.G. Trotter, Covers for regular semigroups and an application to complexity, J. Pure Appl. Algebra 105 (1995), 319-328. MR 96m:20100
  • 40. P.G. Trotter, Relatively free bands of groups, Acta Sci. Math. (Szeged) 53 (1989), 19 - 31. MR 91d:20067
  • 41. P.G. Trotter, E-varieties of regular semigroups, in Semigroups, Automata and Languages, J. Almeida, G.M.S. Gomes and P.V. Silva eds, World Scientific Publ. Co., Singapore, 1994, 247-262.
  • 42. Y.T. Yeh, The existence of e-free objects in e-varieties of regular semigroups, Int. J. Alg. Comp. 2 (1992), 471-484. MR 94d:20085
  • 43. S. Zhang, An infinite order operator on the lattice of varieties of completely regular semigroups, Algebra Universalis 35 (1996), 485-505. MR 97e:20082
  • 44. S. Zhang, Completely regular semigroup varieties generated by Mal'cev products with groups, Semigroup Forum 48 (1994), 180-192. MR 95a:20063

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 20M17, 20M07

Retrieve articles in all journals with MSC (1991): 20M17, 20M07


Additional Information

Peter R. Jones
Affiliation: Department of Mathematics, Statistics and Computer Science, Marquette University, P.O. Box 1881, Milwaukee, Wisconsin 53201-1881
Email: jones@mscs.mu.edu

Peter G. Trotter
Affiliation: Department of Mathematics, University of Tasmania, Hobart, Australia, 7001
Email: trotter@hilbert.maths.utas.edu.au

DOI: https://doi.org/10.1090/S0002-9947-97-01638-3
Received by editor(s): August 15, 1994
Additional Notes: The authors are indebted to the Australian Research Council for their support of this research. The first author also gratefully acknowledges the support of National Science Foundation grant INT-8913404.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society