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COMPUTING CENTRALISERS IN [FINITELY GENERATED FREE]-BY-CYCLIC GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 04 June 2024, pp. 1-10
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- Article
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CENTRALISERS IN THE INFINITE SYMMETRIC INVERSE SEMIGROUP
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 87 / Issue 3 / June 2013
- Published online by Cambridge University Press:
- 28 September 2012, pp. 462-479
- Print publication:
- June 2013
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For an arbitrary set
$X$ (finite or infinite), denote by 
$\mathcal {I}(X)$ the symmetric inverse semigroup of partial injective transformations on 
$X$. For 
$ \alpha \in \mathcal {I}(X)$, let 
$C(\alpha )=\{ \beta \in \mathcal {I}(X): \alpha \beta = \beta \alpha \}$ be the centraliser of 
$ \alpha $ in 
$\mathcal {I}(X)$. For an arbitrary 
$ \alpha \in \mathcal {I}(X)$, we characterise the transformations 
$ \beta \in \mathcal {I}(X)$ that belong to 
$C( \alpha )$, describe the regular elements of 
$C(\alpha )$, and establish when 
$C( \alpha )$ is an inverse semigroup and when it is a completely regular semigroup. In the case where 
$\operatorname {dom}( \alpha )=X$, we determine the structure of 
$C(\alpha )$in terms of Green’s relations.
