متجر المتحف مفتوح.

Because Arabic often leaves out the present-tense to be.

So متجر المتحف مفتوح literally looks like museum shop open, but it means The museum shop is open.

This kind of sentence is called a nominal sentence. In the present tense, Arabic usually does not use a separate word for is / are / am.

• متجر المتحف = the museum shop
• مفتوح = open

If you wanted a past or future meaning, then Arabic would use a verb:

• كان متجر المتحف مفتوحًا = The museum shop was open.
• سيكون متجر المتحف مفتوحًا = The museum shop will be open.

Because متجر المتحف is an iḍāfa construction, often called a genitive construction or noun annexation.

In an iḍāfa:

• the first noun usually does not take الـ
• the second noun shows whether the whole phrase is definite or indefinite

So:

• متجر متحف = a museum shop / a shop of a museum
• متجر المتحف = the museum shop / the shop of the museum

Even though متجر has no الـ, the whole phrase is definite because المتحف is definite.

The relationship is possession or association.

So متجر المتحف can mean:

• the museum’s shop
• the shop of the museum
• the museum shop

This is a very common Arabic structure. The first noun is the thing being described, and the second noun tells you what it belongs to or is associated with.

Other examples:

• باب البيت = the door of the house
• مدير المدرسة = the director of the school
• كتاب الطالب = the student’s book

Because in Arabic possession normally goes in the order:

thing possessed + possessor

So Arabic says:

• متجر المتحف = shop the-museum

where English says:

• the museum shop
• or the shop of the museum

This is just the normal Arabic pattern for iḍāfa phrases.

Because مفتوح here is not an adjective inside the noun phrase. It is the predicate of the sentence.

The structure is:

• متجر المتحف = subject
• مفتوح = predicate

In this kind of sentence, the predicate is usually indefinite, so it does not take الـ.

Compare:

• المتحف الكبير = the big museum
  Here الكبير is an adjective describing المتحف, so it takes الـ.

But:

• المتحف كبير = the museum is big
  Here كبير is the predicate, so it does not take الـ.

So in your sentence, مفتوح means is open, not the open.

Because it agrees with متجر, which is a masculine singular noun.

In Arabic, adjectives and predicate adjectives usually agree with the noun in:

• gender
• number
• definiteness when they are true attributive adjectives

Here the important part is gender and number:

• متجر is masculine singular
• so مفتوح is masculine singular

If the noun were feminine, you would expect مفتوحة instead.

For example:

• المكتبة مفتوحة = The library is open.

It is an adjective in form, but in this sentence it functions as the predicate.

So grammatically, it is describing the state of the subject:

• متجر المتحف = the museum shop
• مفتوح = open

Arabic often uses adjectives this way in nominal sentences:

• الجو جميل = The weather is nice.
• الباب مغلق = The door is closed.
• الدرس سهل = The lesson is easy.

So مفتوح is an adjective meaning open, but here it works like is open in English.

With full case endings, it would be:

مَتْجَرُ الْمَتْحَفِ مَفْتُوحٌ

A careful pronunciation would be roughly:

matjaru al-matḥafi maftūḥun

Here is why:

• متجرُ: nominative, because it is the subject
• المتحفِ: genitive, because it is the second part of the iḍāfa
• مفتوحٌ: nominative, because it is the predicate

In normal everyday writing, these short vowels are usually omitted, so you usually just see:

متجر المتحف مفتوح

Yes. Depending on context, it could be rendered as:

• The museum shop is open.
• The shop of the museum is open.
• The museum’s shop is open.

In most everyday contexts, The museum shop is open is the most natural English translation.

You could, but it would mean something a little different.

• متجر المتحف = the museum shop / the shop belonging to the museum
• متجر في المتحف = a shop in the museum

The first one suggests a specific shop associated with the museum itself.
The second one only says that there is a shop located inside the museum.

So متجر المتحف is the better choice if you mean the official museum shop.

Yes, but the basic order here is the most neutral and common:

• متجر المتحف مفتوح

Arabic can sometimes move the predicate for emphasis:

• مفتوحٌ متجرُ المتحفِ

That can be grammatical in formal Arabic, but it is less neutral and more marked. For a learner, the original order is the safest and most standard.

It is the passive participle of the verb فتح meaning to open.

So:

• فتح = to open
• مفتوح = opened / open

In many contexts, passive participles in Arabic work like ordinary adjectives:

• باب مفتوح = an open door
• المحل مفتوح = the shop is open

So even though its form comes from a verb, in this sentence you can simply understand it as the adjective open.