We were shopping at the mall.
It's late and all the shops were starting to close.
"They're closing their shop."
Will the second sentence mean the shop is bankrupt?
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In this sentence closing does not have an object (it's intransitive), and for this meaning "They" can only apply to the shop, not the staff.
They're closing their shop
In this sentence, "They" could refer to the the owners (which might imply that the shop was closing permanently) or to the staff (which would mean that right now the staff are outside the shop and locking the doors). If the owners also work in the shop, if could mean either.
Either sentence could mean that the shop is closing at the end of the day, or that it is closing permanently. Unless the context indicates otherwise, I would assume that the shop was closing at the end of the day, especially if it was late in the day and other shops are closing too.
To indicate that the shop was closing permanently, you could use one of these expressions.
The shop is closing its doors
The shop is closing down
They are closing down
shut up shop used to be common, but has declined in post-war years: more so in the US than in the UK.
They are closing.
Both could either mean that the shop is closed for the night or that it is bankrupt. You'd have to look at the shop to be able to tell.
They are closing the shop = closing for the day
The store is closing down = closing permanently.
Yes, the second sentence (They're closing their shop.) could mean they're going out of business, but it's equally possible it means it's closed just for the day. Same with the first sentence.
But there's an idiom, used in American English only, that the other answers haven't mentioned, which is this:
They're closing up shop.
This usually means the shop is going out of business.
John: Have you been to Orange Julius lately?
Mary: No, Orange Julius has closed up shop.
John: I might cry.
With this idiom, you omit the possessive pronoun "there":
Closed up shop [Correct]
Closed up their shop [Not an idiom]
Here are a few references that support what I'm saying:
For what it's worth, I've noticed in all these references "closed forever" is mentioned first and "closed temporarily" is mentioned second.
Here's a reference to shut up shop, supposedly a British idiom: