Maximal subgroups of the minimal ideal of a free profinite monoid are free. (English) Zbl 1220.20045
The author shows that the maximal subgroup of the minimal ideal of a finitely generated free profinite monoid is a free profinite group. This answers a question asked by S. W. Margolis in the 1990s. More generally, if \(\mathbf H\) is an extension-closed pseudovariety of finite groups that contains a cyclic \(p\)-group for infinitely many primes \(p\) and \(\overline{\mathbf H}\) is the pseudovariety of all finite monoids whose subgroups lie in \(\mathbf H\), then the maximal subgroup of the minimal ideal of a finitely generated free pro-\(\overline{\mathbf H}\) monoid is a free profinite group.
Reviewer’s remark. A further generalization has been obtained by A. Costa and the author [Proc. Lond. Math. Soc. (3) 102, No. 2, 341-369 (2011; Zbl 1257.20054)].
Reviewer’s remark. A further generalization has been obtained by A. Costa and the author [Proc. Lond. Math. Soc. (3) 102, No. 2, 341-369 (2011; Zbl 1257.20054)].
Reviewer: Mikhail Volkov (Ekaterinburg)
MSC:
| 20M05 | Free semigroups, generators and relations, word problems |
| 20E18 | Limits, profinite groups |
| 20M07 | Varieties and pseudovarieties of semigroups |
| 20M12 | Ideal theory for semigroups |
| 20A15 | Applications of logic to group theory |
Keywords:
free profinite monoids; free profinite groups; minimal ideals; extension-closed pseudovarieties; pseudovarieties of finite monoidsCitations:
Zbl 1257.20054References:
| [1] | D. Allen, Jr. and J. Rhodes, Synthesis of classical and modern theory of finite semigroups. Advances in Mathematics 11 (1973), 238–266. · doi:10.1016/0001-8708(73)90010-8 |
| [2] | J. Almeida, Finite semigroups and universal algebra, Volume 3 of Series in Algebra. World Scientific Publishing Co. Inc., River Edge, NJ, 1994. Translated from the 1992 Portuguese original and revised by the author. · Zbl 0844.20039 |
| [3] | J. Almeida, Profinite groups associated with weakly primitive substitutions, Fundamental’naya i Prikladnaya Matematika 11 (2005), 13–48. Translation in Journal of Mathematical Sciences (N.Y.) 144 (2007), 3881–3903. · Zbl 1110.20022 |
| [4] | J. Almeida and M. V. Volkov, Profinite identities for finite semigroups whose subgroups belong to a given pseudovariety, Journsl of Algebra and its Applications 2 (2003), 137–163. · Zbl 1061.20050 · doi:10.1142/S0219498803000519 |
| [5] | J. Almeida and M. V. Volkov. Subword complexity of profinite words and subgroups of free profinite semigroups, International Journal of Algebra and Computation 16 (2006), 221–258. · Zbl 1186.20040 · doi:10.1142/S0218196706002883 |
| [6] | M. Arbib, The automata theory of semigroup embeddings, Journal of the Australian Mathematical Society 8 (1968), 568–570. · Zbl 0159.02502 · doi:10.1017/S1446788700006224 |
| [7] | A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups. Vol. I, Mathematical Surveys, No. 7. American Mathematical Society, Providence, RI, 1961. · Zbl 0111.03403 |
| [8] | S. Eilenberg, Automata, Languages, and Machines. Vol. B, Academic Press, New York, 1976. With two chapters (”Depth decomposition theorem” and ”Complexity of semi-groups and morphisms”) by Bret Tilson, Pure and Applied Mathematics, Vol. 59. |
| [9] | K. Iwasawa, On solvable extensions of algebraic number fields, Annals of Mathematics (2) 58 (1953), 548–572. · Zbl 0051.26602 · doi:10.2307/1969754 |
| [10] | K. Krohn, J. Rhodes and B. Tilson, in Algebraic Theory of Machines, Languages, and Semigroups (Michael A. Arbib, ed.) With a major contribution by Kenneth Krohn and John L. Rhodes. Academic Press, New York, 1968, Chapters 1, 5–9. |
| [11] | S. Margolis, Maximal pseudovarieties of finite monoids and semigroups, Izvestiya Vysshikh Uchebnykh Zavedeniĭ Matematika (1) (1995), 65–70. Translation in Russian Math. (Iz. VUZ) 39 (1995), 60–64. · Zbl 0846.20062 |
| [12] | B. H. Neumann, Embedding theorems for semigroups, Journal of the London Mathematical Society 35 (1960), 184–192. · Zbl 0090.01701 · doi:10.1112/jlms/s1-35.2.184 |
| [13] | J. Rhodes and B. Steinberg, Profinite semigroups, varieties, expansions and the structure of relatively free profinite semigroups, International Journal of Algebra and Computation 11 (2001), 627–672. · Zbl 1026.20057 · doi:10.1142/S0218196701000784 |
| [14] | J. Rhodes and B. Steinberg, Closed subgroups of free profinite monoids are projective profinite groups, The Bulletin of the London Mathematical Society 40 (2008), 375–383. · Zbl 1153.20029 · doi:10.1112/blms/bdn017 |
| [15] | J. Rhodes and B. Steinberg, The q-Theory of Finite Semigroups, Springer Monographs in Mathematics, Springer, New York, 2009. · Zbl 1186.20043 |
| [16] | L. Ribes and P. Zalesskii, Profinite Groups, Volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 2000. |
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