By any reasonable defination, Apple has a monopoly on digital music players. Really.
By any reasonable definition, people who use phrases like "by any reasonable definition" without actually providing an example of a "reasonable definition" appear to be foolish. So here is a "reasonable definition."
In economics, a monopoly (from the Latin word monopolium - Greek language monos, one + polein, to sell) is defined as a persistent market situation where there is only one provider of a product or service. Primary characteristics of a monopoly
Single Seller: For a pure monopoly to take place, only one company can be selling the good. A company can have a monopoly on certain goods and not on other goods.
No close substitutes: Monopoly is not merely the state of having a unique or recognizable product, but also that there are no close substitutes available for the function the good fills.
Price maker: Because a single firm controls the total supply in a pure monopoly, it is able to exert a significant degree of control over the price by changing the quantity supplied.