A partially rented or leased property, or a building that serves dual purposes with incomplete tenancy.
From demi- (half) + premise (a building or part of one). This legal/real estate term emerged in 17th-century England to describe properties with ambiguous ownership or partial occupation rights.
In old London, many buildings were 'demipremises'—a shopkeeper might rent the ground floor while the owner lived above, creating complex legal situations that kept lawyers very busy.
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