This is an introduction to the foundations of economic property rights the- ory (EPRT). In this volume, a first step in the EPRT research program, rules concerning economic property rights (e.p.r.s), entrepreneurial agreements, and enterprises are discussed. Introduced concept of e.p.r.s is an extension of the traditional concept of pairing of residual rights of control and residual rights of returns in the economic theory. Its importance in economics is generated from a general impossibility of making a complete contract, concerning e.p.r.s, for any nontrivial economic transaction. The volume offers a theoretical ex- tension of mathematical economics, applying recent results of Hopf algebras, quasi-Hopf algebras, representation theory, theory of categories, and deforma- tion theories, in looking for suitable mathematical methodology of economic property rights theories and foundations of general theory of economic agree- ments. The idea is to construct a kind of mathematical application in which any fundamental formal entity and/or operation has an empirical economic interpretation. This approach is seen as a way to cope with an extreme com- plexity of economic phenomena under consideration and requests for precise formulation of models where meaningful answers and solutions of problems are only those which are obtained rigorously. The proposed extensions in math- ematical economics and property rights theory are to provide rich enough foundations to follow complexity of economic property rights in the exact way, and to identify where there is an appropriate method providing ade- quate solution, and also to find problems where in general there is no such methodology.