"Of any mall" vs. "of any malls"

From Practical English Usage by Michael Swan:

Any is very often used with uncountable nouns and plural nouns. It can have the same kind of meaning as the indefinite article a/an has with singular countable nouns. E.g.: I haven't got a car, and I haven't got any money to buy one.

Is there a tin-opener in the house? And are there any plates?

With this meaning any is unusual with singular countable nouns.

She hasn't got a job. - NOT: She hasn't got any job.
Do you know a good doctor? - NOT: Do you know any good doctor?

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With an uncountable noun or plural noun, any usually suggests the idea of an indefinite amount or indefinite number. When there is no idea of quantity or number, no article is used.

Is there any water in that can?
Is there water on the moon?

Any can be used to emphasise the idea of 'free choice', with the meaning of 'it doesn't matter who/which/what'. With this meaning, any is common in affirmative clauses as well as questions and negatives, and is often used with singular countable nouns as well as uncountables and plurals."

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I agree with you: I don't find it acceptable to use plural "malls" in "the most comprehensive loyalty rewards program of any malls in the area."

I think the problem with using "malls" is that the preceding noun "program" is singular, and any single program will be associated with only a single mall. It would make sense to say "ABC Mall and XYZ Mall have the most comprehensive loyalty rewards programs of any malls in the area." But when using the singular noun "program", it doesn't seem to work for me to switch to plural in the later part of this noun phrase.

I would say that Jim's suggested rephrasing "the most ... of any of the malls..." works fine because "any of the malls" is valid as a singular noun phrase.

ABC Mall has the most . . . of any OTHER mall in the area; or,

ABC Mall has the most . . . of any OF THE OTHER malls in the area.

The first is the more correct of the two.

To make several (covered) points as clear as possible:   "any" is syntactically singular, can be used with both "count" and "mass" nouns, and invokes doubtful- or non-existence in frame semantics / construction grammar.

One should probably think either "of any other mall" or of "any mall", or "of all malls" + "in the area", but even so, I find it wrong yet: would it not sound logically that

'ABC MALL HAS THE BEST PROGRAM OF A MALL IN THE AREA?'

I am afraid your statement copies the structure of a different kind:

1. This car IS (to be) the strongest (of all cars)(in this area).
2. This car HAS (to have) the strongest engine (of all) (CAR ENGINES). where when one changed OF ALL CAR ENGINES with OF ANY CARS in statement no 2:
3. "This car has the strongest engine of any car(s)", either with plural or singular, it does not look good to me (but then again, I am not a native in English). Should it not be at least OF ANY CAR'S? (if that still sounds normal) I understand TO BE THE BEST OF ANY OF YOUR KIND, but TO HAVE THE BEST (smth.) OF ANY OF YOUR KIND, that I DON'T.

When you use the superlative in a comparison - "the most" (just like "the most of all") - you mean "more than ANY OTHER" or "more than ALL THE OTHERS", and not "more than ANY OTHERS".

I think the problems occurred due to syntax, as "the most comprehensive loyalty reward program" (which already implies thinking "most ___ OF all programs") is followed by an "of" that (re)connects ABC Mall with the rest of the sentence.

Is just like `Peter Pan has the smallest hat of any man in this area', which makes me think that the hat does not even have to belong to him necessarily, he only somehow got it. But is that right to say? I believe that 'OUT OF all malls in the area, ABC Mall has the best ___ " and "In this area, Peter Pan is the man with the smallest hat" work better, but I might be completely wrong.