In the following sentences, why some of the proper nouns are preceded with a definite article, some not? Are all the usages of the definite articles correct in the following sentences?

The memorial is a monument honoring Abraham Lincoln.

is a private university.

He arrived at John F. Kennedy airport.

He visited the President Kennedy Memorial Hall.

The was transformed into the current
I don't think there is a satisfactory answer to your question. The use of "the" with proper nouns is somewhat dependent on arbitrary choices.

Memorials are not distinct enough, perhaps, to allow the dropping of "the". Maybe that's because a memorial can take so many different forms. the Viet Nam memorial, the Lincoln memorial, the World War II memorial

Airports, on the other hand, and most universities, are treated differently. O'Hare airport, John F. Kennedy airport, Indiana University, Michigan State University. Except "the" is used with universities when the first words are "University of". the University of Michigan, the University of Maryland.

He visited the President Kennedy Memorial Hall could probably correctly omit "the", at least to my ear.

Imperial University and National Taiwan University could probably correctly omit "the", but I think "the" is used in your sample sentence above for different reasons. Here you want to say that the X was changed to the Y.
With "of Taiwan", however, it would definitely be the Imperial University of Taiwan.

I understand this way:

1. The Lincoln Memorial takes THE because it is a public institution.
(EX) The British Museum, The Bodelian Library, but Carnegie Hall.
2. Airport names always don’t take THE.

Names of universities and companies have almost no rules for the requirement of THE. But generally speaking, the form of "university of a geographical name" takes THE. "The Imperial University" in your sentence was formally "The Imperial University of Taipei".

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Thank you very much for the reply, CJ.
Many thanks to you too, Paco.

Can you tell me the cases where the name s are distinct enough to allow the dropping of "the" as you said that Memorials are not distinct enough?
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Caljim is right. We can't answer every question based the rules of the book. Sometimes it's a matter of trend, sometimes it's individual preception of the question but most importantly it's the context in which the question comes from.

The Lincoln memorial is a monument honoring Abraham Lincoln.

In E1, the article “the” is really definitive to “monument”, not Lincoln Memorial. Therefore, “the “ is required as a "definer".

Lincoln Memorial University is a private university.

Lincoln Memorial Univ. is a complete noun. It’s like saying “my car is red” vs. “the red car is mine”.

He arrived at [the]John F. Kennedy airport.- This one actually needs the article “the” because it’s a proper name.
Hi, Goodman!

Just a little FYI: I would not personally use "the" with John F. Kennedy Airport, or with any other airport, even if it were named after a person.

Thanks, CJ.

Thanks for your advice. I'll gracefully take it into consideration.

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