The structure of centralizers in the finite symmetric inverse semigroup
Janusz Konieczny · Communications in Algebra · 2020
For an integer n≥1, let In be the symmetric inverse semigroup of partial injective transformations on an n-element set. For λ∈In, denote by C(λ) and C2(λ), respectively, the first and second centralizer of λ in In. We determine the structure of C(λ) and C2(λ) in terms of Green’s relations, including the partial order of J-classes, and express C2(λ) as a direct product of cyclic groups with zero adjoined and a monogenic monoid. For each individual Green relation G, we determine λ∈In such that G in C(λ) is inherited from In, and λ such that all Green relations in C(λ) are inherited from In. We also provide a representation of the partial order of J-classes in C2(λ) as a known lattice, and describe λ∈In such that C2(λ)=C(λ), which gives a class of maximal commutative subsemigroups of In.