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Ivanov, S. V.; Margolis, S. W.; Meakin, J. C.

On one-relator monoids and one-relator groups. (English) Zbl 0985.20021

J. Pure Appl. Algebra 159, No. 1, 83-111 (2001).
The present paper is concerned with the word problem for one-relator inverse monoids with a presentation of the form \(M=\text{Inv}\langle A\parallel w=1\rangle\) where \(w\) is some word in \(A\cup A^{-1}\). The authors establish the following two main results: If \(w\) is a cyclically reduced word then (1) the word problem for \(M\) is decidable if the membership problem for \(P_w\) (a certain submonoid of \(G=\text{Gp}\langle A\parallel w=1\rangle\) called the prefix of \(G\)) in \(G\) is decidable (Theorem 3.1), and (2) \(M\) is \(E\)-unitary (Theorem 4.1), that is the natural morphism from \(M\) onto its maximal group image is idempotent-pure. The second result solves a conjecture due to Margolis, Meakin and Stephen published in 1986. The paper is concluded by applying the fact that \(M\) is \(E\)-unitary to solve the word problem for one-relator inverse monoids in certain cases.
Reviewer: C.G.Chehata (Alexandria)

Cited in 1 Review
Cited in 17 Documents

MSC:

20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20M05 Free semigroups, generators and relations, word problems
20F05 Generators, relations, and presentations of groups
20M18 Inverse semigroups

Keywords:

word problem; one-relator inverse monoids; presentations; cyclically reduced words; membership problem
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References:

[1] S.I. Adjan, Defining relations and algorithmic problems for groups and semigroups, Trudy Math. Inst. Steklov 85 (1966).; S.I. Adjan, Defining relations and algorithmic problems for groups and semigroups, Trudy Math. Inst. Steklov 85 (1966). · Zbl 0204.01702
[2] Adjan, S. I.; Oganessian, J., Problems of equality and divisibility in semigroups with a single defining relation, Mathem. Zametki, 41, 412-421 (1987) · Zbl 0617.20035
[3] Benois, M., Parties rationnelles du groups libre, C.R. Acad. Sci. Paris, Sér. A, 269, 1188-1190 (1969) · Zbl 0214.03903
[4] A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vol. 1, 1961, Vol. 2, 1967, American Mathematical Society, Providence, RI.; A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vol. 1, 1961, Vol. 2, 1967, American Mathematical Society, Providence, RI. · Zbl 0111.03403
[5] V. Guba, On the relationship between the word problem and the divisibility problem in one relator semigroups, Russian Math. Izv. 61, 1997, in preparation.; V. Guba, On the relationship between the word problem and the divisibility problem in one relator semigroups, Russian Math. Izv. 61, 1997, in preparation. · Zbl 0934.20037
[6] Ivanov, S. V., The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput., 4, 1-308 (1994) · Zbl 0822.20044
[7] Ivanov, S. V.; Schupp, P. E., On the hyperbolicity of small cancellation groups and one-relator groups, Trans. Amer. Math. Soc., 350, 1851-1894 (1998) · Zbl 0915.20016
[8] S.V. Ivanov, J.C. Meakin, On asphericity and the Freiheitssatz for certain finitely presented groups, J. Pure Appl. Algebra 159 (2001) 121-129, this issue.; S.V. Ivanov, J.C. Meakin, On asphericity and the Freiheitssatz for certain finitely presented groups, J. Pure Appl. Algebra 159 (2001) 121-129, this issue. · Zbl 0996.20019
[9] Lallement, G., On monoids presented by a single relation, J. Algebra, 32, 370-388 (1977) · Zbl 0307.20034
[10] Lallement, G., Semigroups and Combinatorial Applications (1979), Wiley: Wiley New York · Zbl 0421.20025
[11] Lyndon, R. C.; Schupp, P. E., Combinatorial Group Theory (1977), Springer: Springer Berlin · Zbl 0368.20023
[12] Magnus, W., Das Identitätsproblem für Gruppen mit einer definierenden Relation, Math. Ann., 105, 1931 (5274)
[13] W. Magnus, J. Karras, D. Solitar, Combinatorial Group Theory, Wiley Interscience, New York, 1966.; W. Magnus, J. Karras, D. Solitar, Combinatorial Group Theory, Wiley Interscience, New York, 1966. · Zbl 0138.25604
[14] Margolis, S.; Meakin, J., Inverse monoids, trees and context-free languages, Trans. Amer. Math. Soc., 335, 259-276 (1993) · Zbl 0795.20043
[15] Margolis, S. W.; Meakin, J.; Stephen, J. B., Some decision problems for inverse monoid presentations, (Goberstein, S.; Higgins, P., Semigroups and Their Applications (1986), Reidel: Reidel Dordrecht), 99-110 · Zbl 0622.20046
[16] Munn, W. D., Free inverse semigroups, Proc. London Math. Soc., 29, 3, 385-404 (1974) · Zbl 0305.20033
[17] Newman, B. B., Some results on one-relator groups, Bull. Amer. Math. Soc., 74, 568-571 (1968) · Zbl 0174.04603
[18] A.Yu. Ol’shanskii, Geometry of Defining Relations in Groups, Nauka, Moscow, 1989 (English translation in: Math and Its Applications (Soviet series), Vol. 70, Kluwer, Dordrecht, 1991).; A.Yu. Ol’shanskii, Geometry of Defining Relations in Groups, Nauka, Moscow, 1989 (English translation in: Math and Its Applications (Soviet series), Vol. 70, Kluwer, Dordrecht, 1991). · Zbl 0676.20014
[19] Petrich, M., Inverse Semigroups (1984), Wiley: Wiley New York · Zbl 0546.20053
[20] Schupp, P. E., A strengthened Freiheitssatz, Math. Ann., 221, 73-80 (1976) · Zbl 0307.20019
[21] Silva, P. V., Rational languages and inverse monoid presentations, Internat. J. Algebra Comput., 2, 2, 187-207 (1992) · Zbl 0777.20023
[22] Stephen, J. B., Presentations of inverse monoids, J. Pure Appl. Algebra, 63, 81-112 (1990) · Zbl 0691.20044
[23] Stephen, J. B., Inverse monoids and rational subsets of related groups, Semigroup Forum, 46, 1, 98-108 (1993) · Zbl 0807.20048
[24] Watier, G., On the word problem for single relation monoids with an unbordered relator, Internat. J. Algebra Comput., 7, 6, 749-770 (1997) · Zbl 0893.08001
[25] Zhang, L., A short proof of a theorem of Adjan, Proc. AMS, 116, 1-3 (1992) · Zbl 0758.20015
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