1. An electronic monetary system comprising: an issuing bank having an on-line accounting system; electronic representations of currency that are credited in said on-line accounting system as current liabilities of said issuing bank; a money generator module associated with said issuing bank, for generating said electronic representations of currency; a teller module associated with said issuing bank, capable of storing said electronic representations of currency, and intermediating banking transactions involving said electronic representations of currency; a transaction module capable of storing said electronic representations of currency, performing on-line transactions with said issuing bank, and exchanging said electronic representations of currency with other transaction modules in off-line transactions; where said electronic representations of currency each include an original monetary value generated by said money generator module; and said teller and transaction modules having processors operative, when said modules are functioning as transferor modules transferring one of said electronic representations of currency to a transferee module, to generate and include in said transferred electronic representation of currency a transfer record having a transferred monetary value.