Algebra-coalgebra structures and bialgebras

Abstract

A recent paper by Hennicker and Kurz[4] gives a formulation of algebraic operations which respect behavioural equivalence as a pair of an algebra and a coalgebra structures on the same carrier set. The conditions on such a pair are simple but elegant and intuitive. Another elegant formulation, given by Turi and Plotkin[18], uses a similar pair of an algebra and a coalgebra structures, which fits in a certain diagram that involves a natural transformation with distributive property. The paper investigates a relationship between these two similar formulations, and shows when they can be interchangeable and when they cannot.