On one-relator inverse monoids and one-relator groups

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

Research output: Contribution to journalArticlepeer-review


It is known that the word problem for one-relator groups and for one-relator monoids of the form Mon〈A||w=1〉 is decidable. However, the question of decidability of the word problem for general one-relation monoids of the form M=Mon〈A||u=v〉 where u and v are arbitrary (positive) words in A remains open. The present paper is concerned with one-relator inverse monoids with a presentation of the form M=Inv〈A||w=1〉 where w is some word in A∪A-1. We show that a positive solution to the word problem for such monoids for all reduced words w would imply a positive solution to the word problem for all one-relation monoids. We prove a conjecture of Margolis, Meakin and Stephen by showing that every inverse monoid of the form M=Inv〈A||w=1〉, where w is cyclically reduced, must be E-unitary. As a consequence the word problem for such an inverse monoid is reduced to the membership problem for the submonoid of the corresponding one-relator group G=Gp〈A||w=1〉 generated by the prefixes of the cyclically reduced word w. This enables us to solve the word problem for inverse monoids of this type in certain cases.

Original languageEnglish (US)
Pages (from-to)83-111
Number of pages29
JournalJournal of Pure and Applied Algebra
Issue number1
StatePublished - May 8 2001


  • 20F06
  • 20F32
  • 20M05
  • 20M18
  • Primary 20F05
  • Secondary 57M20

ASJC Scopus subject areas

  • Algebra and Number Theory


Dive into the research topics of 'On one-relator inverse monoids and one-relator groups'. Together they form a unique fingerprint.

Cite this