Finite presentability of HNN extensions of inverse semigroups

Erzsebet Dombi, N.D. Gilbert, Nik Ruskuc

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)423-436
Number of pages14
JournalInternational Journal of Algebra and Computation
Volume15
Issue number3
Publication statusPublished - Jun 2005

Fingerprint

HNN Extension
Inverse Semigroup
Finitely Generated
Order Ideal
Necessary
Preorder
Groupoids
Isomorphism
Isomorphic
If and only if

Keywords

  • inverse semigroup
  • finite presentation
  • HNN extension
  • presentability

Cite this

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

In: International Journal of Algebra and Computation, Vol. 15, No. 3, 06.2005, p. 423-436.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Finite presentability of HNN extensions of inverse semigroups

AU - Dombi, Erzsebet

AU - Gilbert, N.D.

AU - Ruskuc, Nik

PY - 2005/6

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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

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