The action of the automorphism group of a finite graded partially ordered set on levels

The inequalities between the numbers of orbits of action of the automorphism group of a finite graded partially ordered set on the levels are obtained. The proofs are based on the notion of $\Delta$-reduced algebras of binary functions introduced in the paper. These algebras are analogies of the $\Delta$-reduced incidence algebras of finite partially ordered sets.