Moreover, if P is a binomial poset then [[summation].sup.*](P) is a Sheffer poset with the factorial function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for n [greater than or equal to] 2.

The boolean lattice [B.sub.n] of rank n is an Eulerian binomial poset with factorial function B(k) = k!

The butterfly poset [T.sub.n] of rank n is an Eulerian binomial poset with factorial function B(k) = [2.sup.k-1] for 1 [less than or equal to] k [less than or equal to] n and atom function A(k) = 2, for 2 [less than or equal to] k [less than or equal to] n, and A(1) = 1.

It is not hard to see that in any n-interval of an Eulerian binomial poset P with factorial function B(k) for 1 [less than or equal to] k [less than or equal to] n, the Euler-Poincare relation is stated as follows:

Then the poset P and its factorial function B(n) satisfy the following conditions:

In Section 3, we show that our new result is appropriate for such a purpose and we illustrate its utility discussing the meaning of a few concrete denotational specifications, among them, the denotational specification of the factorial function and a while-loop.

In order to prove that the meaning of specification (33) is the entire factorial function, we take [summation] = N and consider the Baire partial metric space ([[summation].sub.[infinity]], [p.sub.B]) (cf.

It is evident that [[psi].sup.n.sub.fact]([x.sub.0]) models the behavior of the recursive program that computes the factorial function when the input is exactly n [member of] N.

Therefore, Theorem 2 provides the entire factorial function, the meaning of (33), as the unique fixed point of the mapping [[psi].sub.fact] with infinite length.

[5] Felice Russo, Five Properties of the Smarandache double

factorial function, Smarandache Notions Journal, 13(2002), 103-105.

In this paper, we study the hybrid mean value of the Smarandache triple

factorial function and the Mangoldt function, and give a sharp asymptotic formula.

In reference [2], Professor Jozsef Sandor defined the following analogue of Smarandance double

factorial function as: