No abstract

An important well-known result of Rota describes the relationship between the Mo bius functions of two posets related by a Galois connection. We present an analogous result relating the antipodes of the corresponding incidence Hopf algebras, from which the classical formula can be deduced. To motivate the derivation of this more general result, we first observe that a simple conceptual proof of Rota's classical formula can be obtained by interpreting it in terms of bimodules over the incidence algebras. Bimodules correct the apparent lack of functoriality of incidence algebras with respect to monotone maps. The theory of incidence Hopf algebras is reviewed from scratch and centered around the notion of cartesian posets. Also, the universal multiplicative function on a poset is constructed and an analog for antipodes of the classical Mo bius inversion formula is presented. Academic Press

In this article -that has also the intention to survey some known results in the theory of compact quantum groups using methods different from the standard and with a strong algebraic flavor-we consider compact•-coalgebras and Hopf algebras. In the case of a •-Hopf algebra we present a proof of the characterization of the compactness in terms of the existence of a positive definite integral, and use our methods to give an elementary proof of the uniqueness -up to conjugation by an automorphism of Hopf algebrasof the compact involution appearing in [4]. We study the basic properties of the positive square root of the antipode square that is a Hopf algebra automorphism that we call the positive antipode. We use it -as well as the unitary antipode and Nakayama automorphism-in order to enhance our understanding of the antipode itself.

Abstract. We consider issues related to the origins, sources and initial motivations of the theory of Hopf algebras. We consider the two main sources of primeval development: algebraic topology and algebraic group theory. Hopf algebras are named from the work of Heinz Hopf in the 1940's. In this note we trace the infancy of the subject back to papers from the 40's, 50's and 60's in the two areas mentioned above. Many times we just describe -and/or transcribe parts of -some of the relevant original papers on the subject.

Abstract. We consider different classes of combinatory structures related to Krivine realizability. We show, in the precise sense that they give rise to the same class of triposes, that they are equivalent for the purpose of modeling higher-order logic. We center our attentions in the role of a special kind of Ordered Combinatory Algebras-that we call the Krivine ordered combinatory algebras ( K OCAs)-that we propose as the foundational pillars for the categorical perspective of Krivine's classical realizability as presented by Streicher in [23].Our procedure is the following: we show that each of the considered combinatory structures gives rise to an indexed preorder, and describe a way to transform the different structures into each other that preserves the associated indexed preorders up to equivalence. Since all structures give rise to the same indexed preorders, we only prove that they are triposes once: for the class of K OCA s. We finish showing that in K OCA s, one can define realizability in every higher-order language and in particular in higher-order arithmetic.