### Abstract

Original language | English |
---|---|

Pages (from-to) | 423-436 |

Number of pages | 14 |

Journal | International Journal of Algebra and Computation |

Volume | 15 |

Issue number | 3 |

Publication status | Published - Jun 2005 |

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### Keywords

- inverse semigroup
- finite presentation
- HNN extension
- presentability

### Cite this

*International Journal of Algebra and Computation*,

*15*(3), 423-436.

}

*International Journal of Algebra and Computation*, vol. 15, no. 3, pp. 423-436.

**Finite presentability of HNN extensions of inverse semigroups.** / Dombi, Erzsebet; Gilbert, N.D.; Ruskuc, Nik.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Finite presentability of HNN extensions of inverse semigroups

AU - Dombi, Erzsebet

AU - Gilbert, N.D.

AU - Ruskuc, Nik

PY - 2005/6

Y1 - 2005/6

N2 - HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely generated or finitely presented base is again finitely generated or finitely presented. Our main results depend upon properties of the -preorder in the associated subsemigroups. Let S be a finitely generated inverse semigroup and let U, V be inverse subsemigroups of S, isomorphic via φ: U → V, that are order ideals in S. We prove that the HNN extension S*U,φ is finitely generated if and only if U is finitely -dominated. If S is finitely presented, we give a necessary and suffcient condition for S*U,φ to be finitely presented. Here, in contrast to the theory of HNN extensions of groups, it is not necessary that U be finitely generated.

AB - HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely generated or finitely presented base is again finitely generated or finitely presented. Our main results depend upon properties of the -preorder in the associated subsemigroups. Let S be a finitely generated inverse semigroup and let U, V be inverse subsemigroups of S, isomorphic via φ: U → V, that are order ideals in S. We prove that the HNN extension S*U,φ is finitely generated if and only if U is finitely -dominated. If S is finitely presented, we give a necessary and suffcient condition for S*U,φ to be finitely presented. Here, in contrast to the theory of HNN extensions of groups, it is not necessary that U be finitely generated.

KW - inverse semigroup

KW - finite presentation

KW - HNN extension

KW - presentability

UR - http://www.worldscientific.com/doi/abs/10.1142/S021819670500227X

M3 - Article

VL - 15

SP - 423

EP - 436

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

SN - 0218-1967

IS - 3

ER -