Semidirect products of regular semigroups

Jones, Peter; Trotter, Peter

doi:10.1090/S0002-9947-97-01638-3

Semidirect products of regular semigroups
HTML articles powered by AMS MathViewer

by Peter R. Jones and Peter G. Trotter PDF

Trans. Amer. Math. Soc. 349 (1997), 4265-4310 Request permission

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

Jorge Almeida, Semidirect products of pseudovarieties from the universal algebraist’s point of view, J. Pure Appl. Algebra 60 (1989), no. 2, 113–128. MR 1020712, DOI 10.1016/0022-4049(89)90124-2
Karl Auinger, The word problem for the bifree combinatorial strict regular semigroup, Math. Proc. Cambridge Philos. Soc. 113 (1993), no. 3, 519–533. MR 1207517, DOI 10.1017/S0305004100076179
Karl Auinger, The bifree locally inverse semigroup on a set, J. Algebra 166 (1994), no. 3, 630–650. MR 1280594, DOI 10.1006/jabr.1994.1169
K. Auinger, On the bifree locally inverse semigroup, J. Algebra 178 (1995), no. 2, 581–613. MR 1359904, DOI 10.1006/jabr.1995.1367
K. Auinger and L. Polák, A multiplication of existence varieties of locally inverse semigroups, preprint.
Samuel Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1974. MR 0530382
T. E. Hall, Some properties of local subsemigroups inherited by larger subsemigroups, Semigroup Forum 25 (1982), no. 1-2, 35–49. MR 663168, DOI 10.1007/BF02573586
T. E. Hall, Identities for existence varieties of regular semigroups, Bull. Austral. Math. Soc. 40 (1989), no. 1, 59–77. MR 1020841, DOI 10.1017/S000497270000349X
T. E. Hall, Regular semigroups: amalgamation and the lattice of existence varieties, Algebra Universalis 28 (1991), no. 1, 79–102. MR 1083823, DOI 10.1007/BF01190413
T. E. Hall, A concept of variety for regular semigroups, Monash Conference on Semigroup Theory (Melbourne, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 101–115. MR 1232677
J. M. Howie, An introduction to semigroup theory, L. M. S. Monographs, No. 7, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0466355
P. R. Jones, Mal′cev products of varieties of completely regular semigroups, J. Austral. Math. Soc. Ser. A 42 (1987), no. 2, 227–246. MR 869748, DOI 10.1017/S1446788700028226
Peter R. Jones, An introduction to existence varieties of regular semigroups, Southeast Asian Bull. Math. 19 (1995), no. 1, 107–118. MR 1323420
P.R. Jones, E-free objects and e-locality of completely regular semigroups, Semigroup Forum 51 (1995), 357–377.
Peter R. Jones and Mária B. Szendrei, Local varieties of completely regular monoids, J. Algebra 150 (1992), no. 1, 1–27. MR 1174884, DOI 10.1016/S0021-8693(05)80045-6
Peter R. Jones and Peter G. Trotter, Locality of DS and associated varieties, J. Pure Appl. Algebra 104 (1995), no. 3, 275–301. MR 1361576, DOI 10.1016/0022-4049(94)00054-9
Jiří Kaďourek, 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), no. 1, 1–55. MR 961825, DOI 10.1007/BF02573217
Jiří Kaďourek and Libor Polák, On the word problem for free completely regular semigroups, Semigroup Forum 34 (1986), no. 2, 127–138. MR 868250, DOI 10.1007/BF02573158
Jiří Kaďourek and Mária B. Szendrei, A new approach in the theory of orthodox semigroups, Semigroup Forum 40 (1990), no. 3, 257–296. MR 1038007, DOI 10.1007/BF02573274
J. Kaďourek and M. Szendrei, On existence varieties of $E$-solid semigroups, Semigroup Forum, to appear.
Stuart W. Margolis, Kernels and expansions: an historical and technical perspective, Monoids and semigroups with applications (Berkeley, CA, 1989) World Sci. Publ., River Edge, NJ, 1991, pp. 3–30. MR 1142365
S. W. Margolis and J.-E. Pin, Inverse semigroups and extensions of groups by semilattices, J. Algebra 110 (1987), no. 2, 277–297. MR 910384, DOI 10.1016/0021-8693(87)90046-9
D. B. McAlister, Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups, J. Austral. Math. Soc. Ser. A 31 (1981), no. 3, 325–336. MR 633441, DOI 10.1017/S1446788700019467
D. B. McAlister, Rees matrix covers for locally inverse semigroups, Trans. Amer. Math. Soc. 277 (1983), no. 2, 727–738. MR 694385, DOI 10.1090/S0002-9947-1983-0694385-3
D. B. McAlister, Rees matrix covers for regular semigroups, J. Algebra 89 (1984), no. 2, 264–279. MR 751144, DOI 10.1016/0021-8693(84)90217-5
William R. Nico, On the regularity of semidirect products, J. Algebra 80 (1983), no. 1, 29–36. MR 690701, DOI 10.1016/0021-8693(83)90015-7
K. S. S. Nambooripad, Structure of regular semigroups. I, Mem. Amer. Math. Soc. 22 (1979), no. 224, vii+119. MR 546362, DOI 10.1090/memo/0224
Hanna Neumann, Varieties of groups, Springer-Verlag New York, Inc., New York, 1967. MR 0215899, DOI 10.1007/978-3-642-88599-0
Francis Pastijn, The lattice of completely regular semigroup varieties, J. Austral. Math. Soc. Ser. A 49 (1990), no. 1, 24–42. MR 1054080, DOI 10.1017/S1446788700030214
F. J. Pastijn and P. G. Trotter, Lattices of completely regular semigroup varieties, Pacific J. Math. 119 (1985), no. 1, 191–214. MR 797024, DOI 10.2140/pjm.1985.119.191
Mario Petrich, Inverse semigroups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 752899
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.
Mária B. Szendrei, Orthogroup bivarieties are bilocal, Semigroups with applications (Oberwolfach, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 114–131. MR 1197850
Mária B. Szendrei, On $E$-unitary covers of orthodox semigroups, Internat. J. Algebra Comput. 3 (1993), no. 3, 317–333. MR 1240388, DOI 10.1142/S0218196793000214
Mária B. Szendrei, The bifree regular $E$-solid semigroups, Semigroup Forum 52 (1996), no. 1, 61–82. Dedicated to the memory of Alfred Hoblitzelle Clifford (New Orleans, LA, 1994). MR 1363530, DOI 10.1007/BF02574082
L. A. Skornjakov, Regularity of the wreath product of monoids, Semigroup Forum 18 (1979), no. 1, 83–86. MR 537665, DOI 10.1007/BF02574177
Bret Tilson, Categories as algebra: an essential ingredient in the theory of monoids, J. Pure Appl. Algebra 48 (1987), no. 1-2, 83–198. MR 915990, DOI 10.1016/0022-4049(87)90108-3
P. G. Trotter, Congruence extensions in regular semigroups, J. Algebra 137 (1991), no. 1, 166–179. MR 1090216, DOI 10.1016/0021-8693(91)90086-N
P. G. Trotter, Covers for regular semigroups and an application to complexity, J. Pure Appl. Algebra 105 (1995), no. 3, 319–328. MR 1367877, DOI 10.1016/0022-4049(94)00151-0
P. G. Trotter, Relatively free bands of groups, Acta Sci. Math. (Szeged) 53 (1989), no. 1-2, 19–31. MR 1018670
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.
Y. T. Yeh, The existence of $e$-free objects in $e$-varieties of regular semigroups, Internat. J. Algebra Comput. 2 (1992), no. 4, 471–484. MR 1189674, DOI 10.1142/S0218196792000281
S. Zhang, An infinite order operator on the lattice of varieties of completely regular semigroups, Algebra Universalis 35 (1996), no. 4, 485–505. MR 1392279, DOI 10.1007/BF01243591
Shu Hua Zhang, Completely regular semigroup varieties generated by Mal′cev products with groups, Semigroup Forum 48 (1994), no. 2, 180–192. MR 1256687, DOI 10.1007/BF02573668