A törött pohár már ki van dobva, ezért nincs az asztalon.

They are the same definite article: a / az = the.

The choice depends on the following sound:

• a before a consonant: a törött pohár
• az before a vowel: az asztalon

So it is not a different word or a different case—just the same article in two forms.

Törött means broken.

In Hungarian, adjectives normally come before the noun, just like in English:

• törött pohár = broken glass / broken cup

So A törött pohár is simply the broken glass.

Also, törött is an adjective form derived from the verb törni (to break).

Yes. The dictionary form is kidobni, meaning to throw out / throw away.

It contains:

• ki = out
• dobni = to throw

In many Hungarian sentence types, the verbal prefix separates from the verb. That is what happens here:

• dictionary form: kidobni
• in this sentence: ki van dobva

So ki is still part of the verb meaning, but the grammar of the sentence causes it to appear separately.

It is a very common Hungarian structure: prefix + van + -va/-ve form.

Here:

• ki = out
• van = is
• dobva = thrown

Together, ki van dobva means something like:

• is thrown out
• has been thrown out
• is already in a thrown-out state

This structure often expresses a resulting state rather than focusing on the action itself.

So the sentence is not mainly about the act of throwing it out, but about the fact that the glass is now already gone / discarded.

Dobva is the -va/-ve form of the verb dobni (to throw).

This form often appears in constructions with van:

• meg van írva = it is written
• ki van dobva = it is thrown out / has been thrown out

It is not the same as a regular English tense. Instead, it helps express that something has reached a certain state.

So dobva by itself is not the whole predicate here; the full predicate is ki van dobva.

Because ki van dobva is a complete predicate built with van.

Hungarian often uses van with the -va/-ve form to express a result state:

• be van zárva = it is closed
• el van törve = it is broken
• ki van dobva = it is thrown out

So van is necessary here. You cannot just say A törött pohár már ki dobva.

Már usually means already.

Here it shows that the action/result has happened by now:

• A törött pohár ki van dobva = The broken glass is thrown out.
• A törött pohár már ki van dobva = The broken glass is already thrown out.

It often gives a sense of change of state or completed situation.

Ezért means therefore, for that reason, or that’s why.

It links the first clause to the second:

• A törött pohár már ki van dobva = The broken glass has already been thrown out
• ezért nincs az asztalon = therefore it is not on the table

So the logic is:

1. it has been thrown out
2. therefore it is not on the table

Because the sentence contains two clauses:

• A törött pohár már ki van dobva
• ezért nincs az asztalon

The comma helps mark the boundary between them and shows the cause-result relationship.

This is very natural punctuation in Hungarian when two related clauses are joined like this.

Because van has a special negative form in this kind of use.

For existence or location, Hungarian uses:

• van = is
• nincs = is not

So:

• az asztalon van = it is on the table
• az asztalon nincs / nincs az asztalon = it is not on the table

You do not normally say nem van here.

Asztal means table.

The ending -on means on (on the surface of something). So:

• asztal = table
• asztalon = on the table

And with the article:

• az asztalon = on the table

This ending is called a case ending. Hungarian often uses case endings where English uses prepositions.

Because Hungarian often leaves out the subject when it is clear from context.

In English, you need:

• therefore it is not on the table

In Hungarian, the subject is understood from the previous clause:

• A törött pohár ... ezért nincs az asztalon

The omitted subject is basically it / the broken glass.

This kind of omission is very common in Hungarian.

Yes. Hungarian word order is flexible, and changes in order often change emphasis rather than the basic meaning.

For example, this sentence is fairly neutral:

• A törött pohár már ki van dobva, ezért nincs az asztalon.

A different order could highlight something else, for example:

• Már ki van dobva a törött pohár...
  This puts more attention on the state already thrown out.

But the original version is very natural if you first want to identify the topic as the broken glass and then say something about it.