Reducibility number Original Research Article

Let P be a poset in a class of posets image. A smallest positive integer r is called reducibility number of P with respect to image if there exists a non-empty subset S of P with image and image. The reducibility numbers for the power set image of an n-set image 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.