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Hi,

For the following proper names, I think it is safe to assume that the article 'the' is pretty much stuck in normal sentential situations, but like 'the Pacific Ocean', when you see proper names as they are represented as in signs, they aren't necessary with an English article, as they should accompany their cole (if I could call it that) representative name part. Why is that?

the Tong-gu Gift Shop -- What is their possible sign name? I think the sign most likely doesn't have the article 'the' in it. Why is that?

the Hotel ZZZ -- What is their likely sign name? Would it have the article in its whole (proper) name? To me, the wouldn't likely be part of its sign name. Why is that?

the ZZZ City Museum -- What would be their likely name? Would it have the article? To me, the wouldn't likely be part of its sign name. Why is that?

the Pacific Ocean -- I think if a person saw the name in a map, he would likely see the name without 'the'. What is the reason?

Sorry another question: Is this kind of phrase almost idiomatic and English articles aren't necessary for these types of phrases?

(saw in an instructions part of a puzzle) I think it had a picture of a port and a starboard.

read the leftover letters from port to starboard. -- I think, if a person who doesn't know any better, it seems safer to resort to using a possessive pronoun before the nouns if unsure (like its port to starboard).

Is it similar in structure to 'day to day'?
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None of the signs would usually include "the". Articles like that usually appear in sentences when discussing the place.

Port and starboard are directions. When facing forward on a ship, port is left and starboard is right. I think the puzzle is just saying to read the letters from left to right.
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It would be up to the business owner to decide how he/she would want the name printed on a sign. Some businesses will include the article 'the', but it seems most chose not to. One possible reason is that businesses are listed alphabetically in directories, and most people don't look up a business in the 't' section under 'the.' If businesses where listed that way, the 't' section would be huge. Aside from that, a sign looks 'cleaner' without the article. But, again, it's up to the business owner to decide how to handle it.
Thank you, GL2 and (of course) Vorpar for your squeaky-clean responses.