## Abstract

We generalize the word problem for groups, the formal

language of all words in the generators that represent the identity,

to inverse monoids. In particular, we introduce the \emph{idempotent

problem}, the formal language of all words representing

idempotents, and investigate how the properties of an inverse monoid

are related to the formal language properties of its idempotent problem.

We show that if an inverse monoid is either E-unitary

or has a finite set of idempotents, then its idempotent problem is

regular if and only if the inverse monoid is finite. We

also give examples of inverse monoids with context-free idempotent

problems, including all Bruck-Reilly extensions of finite groups.

language of all words in the generators that represent the identity,

to inverse monoids. In particular, we introduce the \emph{idempotent

problem}, the formal language of all words representing

idempotents, and investigate how the properties of an inverse monoid

are related to the formal language properties of its idempotent problem.

We show that if an inverse monoid is either E-unitary

or has a finite set of idempotents, then its idempotent problem is

regular if and only if the inverse monoid is finite. We

also give examples of inverse monoids with context-free idempotent

problems, including all Bruck-Reilly extensions of finite groups.

Original language | English |
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Pages (from-to) | 1179-1194 |

Number of pages | 16 |

Journal | International Journal of Algebra and Computation |

Volume | 21 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2011 |

## Keywords

- Inverse semigroup
- idempotent
- formal language