A classification of maximal inverse subsemigroups of the finite symmetric inverse semigroups

Maximal inverse subsemigroups of the finite symmetric inverse semigroup I, on the finite set [n] = {1, 2, ..., n} are described. Further, we investigate maximal inverse subsemigroups of the inverse semigroup V(n, r) = {alpha is an element of I-n : /im alpha/ less than or equal to r} for 1 less than or equal to r less than or equal to n - 1, and completely obtain their classification.