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Green’s relations and regularity in centralizers of permutations. (English) Zbl 0924.20049
The author continues his study of the centralizer, $$C(\sigma)$$, of a partial transformation $$\sigma\in PT_n$$. In this paper $$\sigma$$ is a permutation of the base set and Green’s relations and criteria for regularity of $$C(\sigma)$$ are found. The characterization of the Green’s class for $$\alpha\in C(\sigma)$$ involves the partial transformation of the cycle set of $$\sigma$$ induced by $$\alpha$$ and the cycle length $$\ell(a)$$, $$\ell(b)$$ etc. of the cycles $$a$$, $$b$$ etc. of $$\sigma$$.
For example, $$C(\sigma)$$ is regular if and only if for all cycles $$a,b\in C(\sigma)$$, $$\ell(b)$$ divides $$\ell(a)$$ implies $$\ell(b)=\ell(a)$$. If we strengthen the right hand side of this implication to $$b=a$$, then we have a characterization for $$C(\sigma)$$ to be an inverse semigroup (which must be a semilattice of groups).

##### MSC:
 20M20 Semigroups of transformations, relations, partitions, etc.
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