The Petri Box Calculus (PBC) consists of an algebra of box expressions, and a corresponding algebra of boxes (a class of labelled Petri nets). A compo- sitional semantics provides a translation from box expressions to boxes. The synthesis problem is to provide an algorithmic translation from boxes to box expressions. The axiomatisation problem is to provide a sound and complete axiomatisation for the fragment of the calculus under consideration, which captures a particular notion of equivalence for boxes. There are several alternative ways of defining an equivalence notion for boxes, the strongest one being net isomorphism. In this thesis, the synthesis and axiomatisation problems are investigated for net semantic isomorphism, and a slightly weaker notion of equivalence, called duplication equivalence, which can still be argued to capture a very close structural similarity of con- current systems the boxes are supposed to represent. In this thesis, a structured approach to developing a synthesis algorithm is proposed, and it is shown how this may be used to provide a framework for the production of a sound and complete axiomatisation. This method is used for several different fragments of the Petri Box Calculus, and for gener- ating axiomatisations for both isomorphism and duplication equivalence. In addition, the algorithmic problems of checking equivalence of boxes and box expressions, and generating proofs of equivalence are considered as extensions to the synthesis algorithm.