It is illegal for a company to hold a monopoly on a good or service.
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It is illegal for a company to hold a monopoly on a good or service is false statement.

Monopoly is referred to as situation in the market where there is one and only one player of a particular/bunch of goods or services or both. In the situation of monopoly, the single player can charge the customers as per their wish, as there are no big players in current market. Sometimes, to avoid charging high, government bans company to hold a monopoly. Monopoly can be hold be a company either directly or indirectly.

Company can directly compete and try to take down the other competitors, if any, or directly acquire their company. Or company can indirectly build a very supreme product in such a way that no player is able to compete with them. In some cases, government can ban, but in some cases, even government cannot do anything to remove monopoly. Hence, it is not illegal for a company to hold a monopoly in every case, though it is illegal in few cases, but not in all the goods and services.