ON NILPOTENT SUBSEMIGROUPS OF A FINITE SYMMETRICAL INVERSE SEMIGROUP

1. In this paper we study nilpotent subsemigroups of the symmetric inverse semigroup JS(n) consisting of all partial substitutions of the set N = {1, 2,...,n}. In [1] we established that the nilpotency level of such subsemigroups does not exceed n, and we obtained the description of maximal nilpotent subsemigroups and of maximal subsemigroups with zero multiplication. In this paper, using this description, we give the complete description of all maximal nilpotent semigroups of JS(n) of a given nilpotency level k for k less than or equal to n. We denote by psi the mapping JS(n) --> JS(n), pi --> pi(-1), where pi(-1) is a partial substitution inverse to pi. As is known (see [2]), psi, is an involution, i.e., an anti-isomorphism of order 2. All the concepts and notation we do not define here can be found in [3]. 2. Let angle be a partial order on the set N (we always consider only strict orders), and let k(angle) be the length of the partially ordered set (N, angle). We denote by Mon(angle) the set Mon(angle) = {pi is an element of JS(n) \ a is an element of dom is an element of double right arrow alpha(a) > a}.