# Centralizers of full injective transformations in the symmetric inverse semigroup

Centralizers of full injective transformations in the symmetric inverse semigroup Let $$\mathcal {I}(X)$$ I ( X ) be the symmetric inverse semigroup of partial injective transformations on a set X (finite or infinite). For $$\alpha \in \mathcal {I}(X)$$ α ∈ I ( X ) , let $$C(\alpha )=\{\beta \in \mathcal {I}(X):\alpha \beta =\beta \alpha \}$$ C ( α ) = { β ∈ I ( X ) : α β = β α } be the centralizer of $$\alpha$$ α in $$\mathcal {I}(X)$$ I ( X ) . Consider $$\alpha \in \mathcal {I}(X)$$ α ∈ I ( X ) with $${{\mathrm{dom}}}(\alpha )=X$$ dom ( α ) = X . For each Green relation $$\mathcal {G}$$ G , we determine $$\alpha$$ α such that $$\mathcal {G}$$ G in $$C(\alpha )$$ C ( α ) is the restriction of the corresponding relation in $$\mathcal {I}(X)$$ I ( X ) ; $$\alpha$$ α such that all Green relations in $$C(\alpha )$$ C ( α ) are the restrictions of the corresponding relations in $$\mathcal {I}(X)$$ I ( X ) ; $$\alpha$$ α for which $$\mathcal {D}=\mathcal {J}$$ D = J in $$C(\alpha )$$ C ( α ) ; $$\alpha$$ α for which the partial order of $$\mathcal {J}$$ J -classes in $$C(\alpha )$$ C ( α ) is the restriction of the corresponding partial order in $$\mathcal {I}(X)$$ I ( X ) ; and finally $$\alpha$$ α for which the $$\mathcal {J}$$ J -classes in $$C(\alpha )$$ C ( α ) are totally ordered. The descriptions are in terms of the cycle-ray decomposition of $$\alpha$$ α , which is a generalization of the cycle decomposition of a permutation. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Semigroup Forum Springer Journals

# Centralizers of full injective transformations in the symmetric inverse semigroup

, Volume 96 (3) – Jun 20, 2017
15 pages

/lp/springer_journal/centralizers-of-full-injective-transformations-in-the-symmetric-c088Do0W9j
Publisher
Springer US
Subject
Mathematics; Algebra
ISSN
0037-1912
eISSN
1432-2137
D.O.I.
10.1007/s00233-017-9884-3
Publisher site
See Article on Publisher Site

### Abstract

Let $$\mathcal {I}(X)$$ I ( X ) be the symmetric inverse semigroup of partial injective transformations on a set X (finite or infinite). For $$\alpha \in \mathcal {I}(X)$$ α ∈ I ( X ) , let $$C(\alpha )=\{\beta \in \mathcal {I}(X):\alpha \beta =\beta \alpha \}$$ C ( α ) = { β ∈ I ( X ) : α β = β α } be the centralizer of $$\alpha$$ α in $$\mathcal {I}(X)$$ I ( X ) . Consider $$\alpha \in \mathcal {I}(X)$$ α ∈ I ( X ) with $${{\mathrm{dom}}}(\alpha )=X$$ dom ( α ) = X . For each Green relation $$\mathcal {G}$$ G , we determine $$\alpha$$ α such that $$\mathcal {G}$$ G in $$C(\alpha )$$ C ( α ) is the restriction of the corresponding relation in $$\mathcal {I}(X)$$ I ( X ) ; $$\alpha$$ α such that all Green relations in $$C(\alpha )$$ C ( α ) are the restrictions of the corresponding relations in $$\mathcal {I}(X)$$ I ( X ) ; $$\alpha$$ α for which $$\mathcal {D}=\mathcal {J}$$ D = J in $$C(\alpha )$$ C ( α ) ; $$\alpha$$ α for which the partial order of $$\mathcal {J}$$ J -classes in $$C(\alpha )$$ C ( α ) is the restriction of the corresponding partial order in $$\mathcal {I}(X)$$ I ( X ) ; and finally $$\alpha$$ α for which the $$\mathcal {J}$$ J -classes in $$C(\alpha )$$ C ( α ) are totally ordered. The descriptions are in terms of the cycle-ray decomposition of $$\alpha$$ α , which is a generalization of the cycle decomposition of a permutation.

### Journal

Semigroup ForumSpringer Journals

Published: Jun 20, 2017

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