Centralisers in the Infinite Symmetric Inverse Semigroup

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Digital Object Identifier (DOI)


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Bulletin of the Australian Mathematical Society

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For an arbitrary set X (finite or infinite), denote by I(X) the symmetric inverse semigroup of partial injective transformations on X . For α∈I(X) , let C(α)={β∈I(X):αβ=βα} be the centraliser of α in I(X) . For an arbitrary α∈I(X) , we characterise the transformations β∈I(X) that belong to C(α) , describe the regular elements C(α) , and establish when C(α) is an inverse semigroup and when it is a completely regular semigroup. In the case where dom⁡(α)=X , we determine the structure of C(α) in terms of Green’s relations.


The definitive article is openly available on the website of the Bulletin of the Australian Mathematical Society at: https://doi.org/10.1017/S0004972712000779.

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COPYRIGHT: © 2012 Australian Mathematical Publishing Association Inc.

Published online by Cambridge University Press.