Moscow Mathematical Society

Author(s): A. V. Tishchenko
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 68 (2007).
Journal: Trans. Moscow Math. Soc. 2007, 93-118.
MSC (2000): Primary 20M07; Secondary 20E22
Posted: November 21, 2007
Retrieve article in: PDF

Abstract: A semigroup variety is called a Cross variety if it is finitely based, is generated by a finite semigroup, and has a finite lattice of subvarieties. It is established in which cases the wreath product of two semigroup varieties each of which is an atom of the lattice of semigroup varieties is a Cross variety. Furthermore, for all the pairs of atoms ${\bf U}$ and ${\bf V}$ for which this is possible, either a finite basis of identities for the wreath product ${\bf UwV}$ is given explicitly, a finite semigroup generating this variety is found and the lattice of subvarieties is described, or it is proved that such a finite characterization is impossible.

1.: V. A. Artamonov, Chain varieties of groups, Trudy Sem. Petrovsk. 1978 (1978), no. 3, 3-8. (Russian) MR 0494776 (80m:20027)
2.: A. P. Biryukov, Varieties of idempotent semigroups, Algebra i Logika 9 (1970), 255-273; English transl. in Algebra and Logic 9 (1971), 153-164. MR 0297897 (45:6949)
3.: É. A. Golubov and M. V. Sapir, Varieties of finitely approximable semigroups, Dokl. Akad. Nauk SSSR 247 (1979), 1037-1041; English transl. in Soviet Math. Dokl. 20 (1979), 828-832. MR 0550461 (81d:20050)
4.: É. A. Golubov and M. V. Sapir, Varieties of finitely approximable semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 1982, no. 11, 21-29; English transl. in Sov. Math. 26 (1982), no. 11, 25-36. MR 0687309 (85f:20051)
5.: G. Lallement, Semigroups and combinatorial applications, John Wiley and Sons, New York, 1979. MR 0530552 (81j:20082)
6.: H. Neumann, Varieties of groups, Springer-Verlag, New York, 1967. MR 0215899 (35:6734)
7.: V. V. Rasin, Free completely simple semigroups, Ural. Gos. Univ. Mat. Zap. 11 (1979), no. 3, 140-151. (Russian) MR 0573889 (81i:20079a)
8.: A. V. Tishchenko, A remark on semigroup varieties of finite index, Izv. Vyssh. Uchebn. Zaved. Mat. 1990, no. 7, 79-83; English transl. in Soviet Math. (Iz. VUZ) 34 (1990), no. 7, 92-96. MR 1104152 (92h:20087)
9.: A. V. Tishchenko, When does the monoid wreath product of varieties of semigroups have finite index?, Int. Conf. on Algebra (Barnaul, August 1991), Abstracts of talks, Novosibirsk, 1991, p. 144. (Russian)
10.: A. V. Tishchenko, On different definitions of the wreath product of semigroup varieties, Fundam. Prikl. Mat. 2, no. 2 (1996), 233-249. (Russian) MR 1789007
11.: A. V. Tishchenko, The monoid of semigroup varieties with respect to a wreath product, Uspekhi Mat. Nauk 51 (1996), no. 2, 177-178; English transl. in Russian Math. Surveys 51 (1996), 351-352. MR 1401556
12.: A. V. Tishchenko, Wreath products of varieties, and semi-Archimedean varieties of semigroups, Trudy Mosk. Mat. Obshch. 57 (1996), 218-238; English transl. in Trans. Moscow Math. Soc. 1996, 203-222. MR 1468982 (99d:20101)
13.: A. V. Tishchenko, The wreath product of the atoms of the lattice of semigroup varieties, Uspekhi Mat. Nauk 53 (1998), no. 4, 219-220; English transl. in Russian Math. Surveys 53 (1998), 870-871. MR 1668074
14.: A. V. Tishchenko, On the lattice of subvarieties of the monoid wreath product of varieties, 2-nd Int. Conf. ``Semigroups: theory and applications'' (St.-Petersburg, 1999), Abstracts of talks, St. Petersburg, 1999, pp. 98-99. (Russian)
15.: A. V. Tishchenko, The ordered monoid of semigroup varieties with respect to a wreath product, Fundam. Prikl. Mat. 5 (1999), no. 1, 283-305. (Russian) MR 1799534 (2001k:20125)
16.: L. N. Shevrin and M. V. Volkov, Identities of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 1985, no. 11, 3-47; English transl. in Sov. Math. 29 (1985), no. 11, 1-64. MR 0829099 (87f:20094)
17.: L. N. Shevrin and E. V. Sukhanov, Structural aspects of the theory of varieties of semigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 1989, no. 6, 3-39; English transl. in Soviet Math. (Iz. VUZ) 33 (1989), no. 6, 1-34. MR 1017775 (91a:20071)
18.: A. L. Shmel'kin, Wreath products and varieties of groups, Dokl. Akad. Nauk SSSR 157 (1964), 1063-1065. (Russian) MR 0166262 (29:3539)
19.: A. L. Shmel'kin, Wreath products and varieties of groups, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 149-170. (Russian) MR 0193131 (33:1352)
20.: J. Almeida, Semidirect products of pseudovarieties from the universal algebraist's point of view, J. Pure Appl. Algebra 60 (1989), 113-128. MR 1020712 (91a:20068)
21.: J. Almeida, Semidirectly closed pseudovarieties of locally trivial semigroups, Semigroup Forum 40 (1990), 315-323. MR 1038009 (91a:20070)
22.: J. Almeida, On iterated semidirect products of finite semilattices, J. Algebra 142 (1991), 239-254. MR 1125216 (92f:20061)
23.: S. Eilenberg, Automata, Languages, and Machines. Vol. B, Academic Press, New York, 1976. MR 0530383 (58:26604a)
24.: S. Eilenberg and M. P. Schutzenberger, On pseudovarieties, Advances Math. 19 (1976), 413-418. MR 0401604 (53:5431)
25.: T. Evans, The lattice of semigroup varieties, Semigroup Forum 2 (1971), 1-43. MR 0284528 (44:1753)
26.: G. Higman, Some remarks on varieties of groups, Quart. J. Math. Oxford (2) 10 (1959), 165-178. MR 0113925 (22:4756)
27.: C. Irastorza, Base non finie de variétés, Proc. STACS 85 (Saarbrcken, 1985), Lecture Notes in Comput. Sci. 182, Springer-Verlag, Berlin, 1985, pp. 180-186. MR 0786881 (86j:68068)
28.: Yu. G. Koshelev, Varieties preserved under wreath products, Semigroup Forum 12 (1976), 95-107. MR 0399330 (53:3175)
29.: H. Liebeck, Concerning nilpotent wreath products, Proc. Cambridge Phil. Soc. 58 (1962), 443-451. MR 0139656 (25:3087)
30.: J.-E. Pin, On semidirect products of two finite semilattices, Semigroup Forum 28 (1984), 73-81. MR 0729653 (85b:20049)
31.: V. V. Rasin, On the lattice of varieties of completely simple semigroups, Semigroup Forum 17 (1979), 113-122. MR 0527213 (80b:20073)
32.: V. V. Rasin, On the varieties of Cliffordian semigroups, Semigroup Forum 23 (1981), 201-220. MR 0647112 (83c:20073)
33.: B. Tilson, Categories as algebra: an essential ingredient in the theory of monoids, J. Pure and Appl. Algebra 48 (1987), 83-198. MR 0915990 (90e:20061)
34.: A. V. Tishchenko, Wreath products of atoms of the lattice of semigroup varieties, Int. Conf. ``Semigroups and their applications including semigroup rings'' (St. Petersburg, 1995), Abstracts of talks, St. Petersburg, 1995, pp. 72-73. (Russian)

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2000): 20M07, 20E22

A. V. Tishchenko
Affiliation: Finance Academy at the Government of the Russian Federation and Moscow State Humanitarian Boarding Institute, Moscow, Russia

DOI: 10.1090/S0077-1554-07-00166-5
PII: S 0077-1554(07)00166-5
Keywords: Semigroup, atom, lattice of subvarieties, semigroup variety, wreath product, basis of identities, Cross variety
Posted: November 21, 2007
Copyright of article: Copyright 2007, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1547-738X(e) ISSN: 0077-1554(p)

Previous volume \| Table of contents \| Next volume Recently posted articles \| Most recent volume \| All volumes Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS