Automorfismos multiplicativos e internos de uma álgebra de incidência

Abstract

The main goal of this work is to present different necessary and sufficient conditions for a multiplicative automorphism of an incidence algebra to be inner. Firstly it is verified that a multiplicative automorphism is inner if and only if the element of the incidence algebra associated to that automorphism is fractionary. In the sequence, it is presented a necessary and sufficient condition which considers the Hasse diagram of the poset as a directed graph having weights given by the considered multiplicative automorphism. Finally, it is shown that the quotient group of the multiplicative auto- morphisms of an incidence algebra over the intersection of that group with the inner automorphisms of the algebra is isomorphic to the first cohomology group of the poset of that algebra and some examples are provided