On nilpotent subsemigroups of a finite symmetric inverse semigroup
Let \(N = \{1, 2, \dots, n\}\) and denote by \(JS_n\) the symmetric inverse semigroup of all injective partial selfmaps of \(N\). A partition \(\rho = (N_1, N_2, \dots, N_k)\) into \(k\) nonempty blocks is said to be ordered if the blocks are strictly linearly ordered, that is, if \(N_1 < N_2 < \cdots < N_k\). The ordered \(k\)-tuple \((|N_1|, |N_2|, \dots, |N_k|)\) is said to be the type of the ordered partition \(\rho\) and the type \((n_k, n_{k - 1}, \dots, n_1)\) is said to be opposite to the type \((n_1, n_2, \dots, n_k)\). The authors prove that there is a one-to-one correspondence between the maximal nilpotent subsemigroups of \(JS_n\) of nilpotency level \(k\) and the ordered partitions of \(N\) into \(k\) blocks. They show further that two maximal subsemigroups of \(JS_n\) of nilpotency level \(k \geq 3\) are isomorphic if and only if the types of the corresponding ordered partitions coincide and they are anti- isomorphic if and only if those types are opposite.
scientific article; zbMATH DE number 830723
scientific article