On multiplicative automorphisms of incidence algebras

Let Formula: see text be the incidence algebra of the preordered set Formula: see text over the ring Formula: see text. In the case of a finite connected partially ordered set Formula: see text, we prove that the subgroup of inner multiplicative automorphisms is a direct factor of the group of multiplicative automorphisms of the algebra Formula: see text. As a consequence, we obtain several matching criteria of the subgroup of inner multiplicative automorphisms with the group of multiplicative automorphisms.