Reducibility number

Let P   be a poset in a class of posets PP. A smallest positive integer r is called reducibility number of P   with respect to PP if there exists a non-empty subset S of P   with |S|=r|S|=r and P-S∈PP-S∈P. The reducibility numbers for the power set 2n2n of an n  -set (n⩾2)(n⩾2) with respect to the classes of distributive lattices, modular lattices and Boolean lattices are calculated. Also, it is shown that the reducibility number r of the lattice of all subgroups of a finite group G with respect to the class of all distributive lattices is 1 if and only if the order of G has at most two distinct prime divisors; further if r is a prime number then order of G is divisible by exactly three distinct primes. The class of pseudo-complemented u-posets is shown to be reducible. Deletable elements in semidistributive posets are characterized.