On non-commutative analogues of Hopf algebras arising in non-commutative geometry and quantum field theory

The present dissertation is build around two research topics.; One topic, largely discussed within Chapters 1 and 3 is a proposal of a revised way of constructing Incidence and Reduced Incidence Hopf Algebras associated to partially ordered sets (not necessarily lattices). The proposed construction and terminology aims to clarify the ambiguities and technical problems arising from the interesting literature on this topic (as [JR], [Sch94], [HS89], [FG05]).; The second topic, to which the Chapters 4, 5 and 6 are mainly addressed, is the construction and the study of the Non-Commutative version of the Faa di Bruno Hopf Algebra and its relation to non-commutative power series ([BFK], [AEP]), and to mathematical objects such as the Non-Commutative Connes-Kreimer Hopf Algebra (see [Foi02a], [BF03], [BK]) and Schwinger-Dyson equations (see [BK]). This part of the dissertation is a development of a previously published joint paper with Michael Anchelevich and Edward G. Effros ([AEP]).