The structure of centralizers in the finite symmetric inverse semigroup

For an integer n >= 1,let I-n be the symmetric inverse semigroup of partial injective transformations on an n-element set. For lambda is an element of I-n,denote by C(lambda) and C-2(lambda),respectively, the first and second centralizer of lambda inIn.We determine the structure of C(lambda) and C-2(lambda) in terms of Green's relations, including the partial order of J-classes, and express C-2(lambda)as a direct product of cyclic groups with zero adjoined and a monogenic monoid. For each individual Green relationG,we determine lambda is an element of I-n such that G in C (lambda)is inherited fromIn,and lambda such that all Green relations inC(lambda)are inherited fromIn.We also provide a representation of the partial order of J-classes in C-2(lambda)as a known lattice, and describe lambda is an element of I-n such that C-2(lambda)=C(lambda),which gives a class of maximal commutative subsemigroups of I-n.