المصباح هش، وصندوق الكرتون الممتلئ ثقيل.

In present-tense Arabic nominal sentences, the verb to be is usually omitted.

So:

• المصباح هش = The lamp is fragile
• صندوق الكرتون الممتلئ ثقيل = The full cardboard box is heavy

Arabic does this very naturally in the present. If you wanted the past or future, then Arabic would use a verb such as كان.

Because this is not a noun phrase like the fragile lamp. It is a full sentence:

• المصباح = the lamp
• هش = fragile

So هش is the predicate of the sentence, not just a descriptive adjective inside a noun phrase.

Compare:

• المصباح هش = The lamp is fragile
• المصباح الهش = the fragile lamp

That little difference is very important in Arabic.

Because they are predicates, not definite adjectives attached directly to a noun.

In this sentence:

• المصباح هش = The lamp is fragile
• صندوق الكرتون الممتلئ ثقيل = The full cardboard box is heavy

But if you said:

• المصباح الهش = the fragile lamp
• الصندوق الثقيل = the heavy box

then the adjectives would take ال because they are part of a definite noun phrase.

So:

• without ال → usually predicate in a sentence
• with ال → usually adjective modifying a definite noun

Because صندوق الكرتون is an iḍāfa construction, often called a construct phrase.

In an iḍāfa:

• the first noun does not take ال
• the second noun determines whether the whole phrase is definite

So:

• صندوق = box
• الكرتون = the cardboard / the carton

Together:

• صندوق الكرتون = literally box of the cardboard/carton
• in context, something like the cardboard box

Even though صندوق has no ال, the whole phrase is definite because the second noun is definite.

Because it is describing a definite noun phrase, and Arabic adjectives agree with what they describe in definiteness.

Since صندوق الكرتون is definite, the adjective must also be definite:

• صندوق الكرتون الممتلئ = the full cardboard box

If the noun phrase were indefinite, the adjective would also be indefinite:

• صندوق كرتون ممتلئ = a full cardboard box

So the ال on الممتلئ matches the definiteness of the noun phrase it describes.

In context, it is most naturally understood as describing the whole phrase, especially the box: the full cardboard box.

However, Arabic can sometimes be structurally ambiguous here, because:

• صندوق الكرتون is masculine singular
• الكرتون is also masculine singular

So الممتلئ could theoretically be read as matching either one.

In real use, meaning usually makes it clear. A full box is much more natural than full cardboard in this sentence, so learners normally understand it as modifying صندوق الكرتون as a whole.

Because صندوق is a masculine noun in Arabic.

So the predicate is masculine singular:

• صندوق ... ثقيل = The box ... is heavy

Likewise:

• مصباح is also masculine
• so المصباح هش uses a masculine form as well

Arabic adjectives and predicates must agree with the noun in gender and number.

Literally, it is something like box of cardboard/carton.

That is understandable, and Arabic often uses noun-plus-noun structures like this. But depending on context, other expressions may sound more explicit or more natural for cardboard box, such as:

• صندوق من الكرتون = a box made of cardboard
• صندوق كرتوني = a cardboard box

So the phrase in your sentence is understandable, but learners should know that Arabic has more than one way to express this idea.

Because short connecting words in Arabic are often written attached to the following word.

So:

• و = and
• وصندوق = and a box / and the box phrase

This is completely normal Arabic spelling. The و is still a separate word grammatically, even though it is attached in writing.

The same thing happens with other particles too.

With full case endings, it would be:

المصباحُ هشٌّ، وصندوقُ الكرتونِ الممتلئُ ثقيلٌ.

A quick breakdown:

• المصباحُ → nominative, because it is the subject/topic of the nominal sentence
• هشٌّ → nominative, because it is the predicate
• صندوقُ → nominative, because it is the subject/topic of the second nominal sentence
• الكرتونِ → genitive, because it is the second term of the iḍāfa
• الممتلئُ → nominative, because it describes صندوق / the whole noun phrase in nominative position
• ثقيلٌ → nominative, because it is the predicate

In everyday printed Arabic, these endings are usually omitted, but in MSA grammar they are still important.