The structure of centralizers in the finite symmetric inverse semigroup

Conjugacy in inverse semigroups

“…, x k−1 }), while the latter fixes every such x. The following decomposition was proved in [12,Prop. 2.4].…”

“…Cycle-chain-ray decomposition of elements of I(X). The cycle decomposition of a permutation can be extended to the cycle-chain-ray decomposition of a partial injective transformation (see [12]).…”

“…(11) If S is an inverse semigroup, then S is a Clifford semigroup if and only if no two different idempotents of S are conjugate (Theorem 6.8). (12) In the bicyclic monoid B, i-conjugacy coincides with the minimal group congruence (Theorem 7.1). (13) An inverse semigroup S is stable if and only if i-conjugacy and the natural partial order intersect trivially as relations in S × S (Theorem 7.3).…”