This paper presents mathematical and general models of electronic money systems. The goal of the paper is to propose a first framework in which various kinds of e-money systems can be uniformly represented and their security properties can be evaluated and compared. We introduce two kinds of e-money system models; a note-type e-money system model and a balance-type e-money system model. We show that any balance-type e-money system with efficient data transmission cannot be simulated by any note-type e-money system. This implies that balance-type e-money systems are strictly faster in data communication. Then, we show that a forged monetary value can be detected in some note-type e-money systems, while it cannot be detected in any balance-type e-money systems with efficient data communication. This implies that note-type e-money systems seem to be more secure.