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.
Semigroup Forum – Springer Journals
Published: Jun 20, 2017
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