Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T14:39:57.379Z Has data issue: false hasContentIssue false
  • English
  • Français

The word problem for one-relator semigroups

Published online by Cambridge University Press:  24 October 2008

James Howie  and
Stephen J. Pride
James Howie
Affiliation:
Department of Mathematics, University of Glasgow
Stephen J. Pride
Affiliation:
Department of Mathematics, University of Glasgow
Get access Rights & Permissions [Opens in a new window]

Extract

Diagrams have been used in group theory by numerous authors, and have led to significant results (see [4] and the references cited there). The idea of applying diagrams to semigroups seems to be more recent [3, 7, 8]. In the present paper we discuss semi group diagrams and use them to obtain results concerning the word problem for one-relator semigroups. The word problem for one-relator groups has been solved by Magnus [6], but the analogous question for semigroups remains open. We are not able to solve the problem in full generality, but have obtained some partial results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 1 , January 1986 , pp. 33 - 44
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adyan, S. I.. On the embeddability of semigroups in groups. Dokl. Akad. Nauk 133 (1960), 255–257; Soviet Math. Dokl. 1 (1960), 819–821.Google Scholar
[2]Adyan, S. I.. Defining relations and algorithmic problems for groups and semigroups. Proc. Steklov. Inst. Math. 85 (1966).Google Scholar
[3]Krstić, S.. Embedding semigroups in groups. A geometrical approach (Preprint 1984).Google Scholar
[4]Lyndon, R. C.. Problems in combinatorial group theory. Conference Proceedings, Alta, Utah 1984. Ann. of Math. Studies, to appear.Google Scholar
[5]Lyndon, R. C. and Schützenberger, M. P.. The equation aM = bN cP in a free group. Michigan Math. J. 9 (1962), 289–298.CrossRefGoogle Scholar
[6]Magnus, W.. Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann. 106 (1932), 295–307.CrossRefGoogle Scholar
[7]Remmers, J. H.. On the geometry of semigroup presentations. Adv. in Math. 36 (1980),283–296.CrossRefGoogle Scholar
[8]Stallincs, J.. A graph-theoretic lemma and group embeddings. Conference Proceedings, Alta, Utah 1984. Ann. of Math. Studies, to appear.Google Scholar
[9]Whitehead, J. H. C.. On certain elements in a free group. Proc. London Math. Soc. 41 (1936), 48–56.Google Scholar
[10]Whitehead, J. H. C.. On equivalent sets of elements in a free group. Ann. of Math. 37 (1936), 782–800.CrossRefGoogle Scholar
[11]Adyan, S. I.. Word transformations in a semigroup that is given by a system of defining relations. Algebra i Logika 15 (1976), 611–621; Algebra and Logic 15 (1976), 379–386.Google Scholar
[12]Adyan, S. I. and Oganesyan, G. U.. On the word and divisibility problems in semigroups with a single defining relation. Izv. Akad. Nauk SSSR Ser. Mat. 43 (1978), 219–225; Math. USSR Izv. 12 (1978), 207–212.Google Scholar
[13]Book, R. V. and Squier, C.. Almost all one-rule Thue systems have decidable word problems. Discrete Math. 49 (1984), 237–240.CrossRefGoogle Scholar
[14]Jackson, D. A.. Some one-relator semigroup presentations with solvable word problems. Math. Proc. Cambridge Philos. Soc. 99 (1986), to appear.CrossRefGoogle Scholar
[15]Lallement, G.. On monoids presented by a single relation. J. Algebra 32 (1974), 370–388.CrossRefGoogle Scholar
[16]Oganesyan, G. U.. On the problems of equality and divisibility of words in a semigroup with a defining relation of the form a = bA. Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 602–612Google Scholar
Oganesyan, G. U.. On the problems of equality and divisibility of words in a semigroup with a defining relation of the form a = bA. Math. USSR Izv. 12 (1978), 557–566.Google Scholar
[17]Oganesyan, G. U.. On semigroups with a single relation and semigroups without cycles. Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 88–94Google Scholar
Oganesyan, G. U.. On semigroups with a single relation and semigroups without cycles. Math. USSR Izv. 20 (1983), 89–95.CrossRefGoogle Scholar
[18]Oganesyan, G. U.. Isomorphism problem for semigroups with one defining relation. Mat. Zam. 35 (1984), 685–690Google Scholar
Oganesyan, G. U.. Isomorphism problem for semigroups with one defining relation. Math. Notes 35 (1984), 360–363.CrossRefGoogle Scholar
[19]Sarkisyan, O. A.. Word and divisibility problems in semigroups and groups without cycles. Izv. Akad. SSSR Ser. Mat. 45 (1981), 1424–1441Google Scholar
Sarkisyan, O. A.. Word and divisibility problems in semigroups and groups without cycles. Math. USSR Izv. 19 (1982), 643–656.Google Scholar