Can we say cause/causes behind the world is/are eternal?

Nibbana is an ultimate reality, but the 'world' is a concept. What you call the world is merely seeing, hearing, smelling, tasting, feeling, thinking. These are just experiences. There is no arising of experiences in Nibbana.

If we say world has a beginning

But we cannot say this as it has no basis in reality. So any conclusions or questions you come up with based on this assumption will only take you away from the reality. It is not in the spirit of Buddhism to entertain such questions.

Be mindful of the questions arising in the mind that only serve to please your imagination. They will lead you astray from the Buddhist path.

Actually this sort of questions are out of our understanding level, so LORD BUDDHA said not follow these questions, it'll only consume your valuable time and rare chance of being human.

You are still having the trouble with Nirvana, Nirvana is not anything to existed nor not existed, it is a ultimate result for following eight noble path and understand the Four Noble truths. So those who attained Nirvana can not explain to ordinary people,exact explanation, because we can not understand it by our present knowledge Nirvana can not compare with anything. Though as a ordinary human being we can not understand it until we attained Nirvana. Only one way to find out answers for your answers, be faithful and follow the path attained Nirvana.

There is one reality which is eternal and that is Nirvana.

Buddhism does not teach this. AN 3.136 and SN 12.20 say the Laws of Nature is also eternal reality.

Nirvana and the world are closely linked up.

Buddhism does not teach this.

The one who is in the world can attain Nirvana.

This does not mean the world and Nibbana are closely linked.

If we say world has a beginning then Nirvana predates that beginning.

This is speculation. The Buddha was concerned with the "world" of "suffering" (SN 12.44). The world of suffering begins with ignorance.

Only Nirvana existed and then world came into existence. Which means World came from Nirvana as world

This is theism, as taught in the Bible and Brahmanism. It is not Buddhism. Buddhism teaches the world arises from ignorance (SN 12.44).

Which means world must have come out from state of Nirvana. But that is impossible.

No.

Therefore can we at least conclude that cause or causes behind the world are without beginning?

The Buddha taught the beginning of the world is ignorance.

If we say world has a beginning

It's not clear that's true (i.e. perhaps that's not true).

There's a sutta -- Assu Sutta: Tears (SN 15.3) -- which begins:

At Savatthi. There the Blessed One said: "From an inconstruable beginning comes transmigration. A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating & wandering on. What do you think, monks: Which is greater, the tears you have shed while transmigrating & wandering this long, long time — crying & weeping from being joined with what is displeasing, being separated from what is pleasing — or the water in the four great oceans?"

The question of whether "the world" is eternal is one of the unanswered questions.

Part of the ambiguity is, what do you mean by "world":

• The entire universe
• Each and every compound thing within the universe

The latter (compound things) are by definition impermanent -- each thing breaks up and becomes other things -- but it's not clear you can say the same about the collection-of-all-things.

Some Buddhist doctrines (suttas and so on) talk about cosmology or cosmologies, suggesting that they're cyclic or infinite ... so:

• I don't think there's canonical justification for saying "the world has a beginning"
• I don't think it's knowable by personal experience
• I think that trying to deduce it using logic alone is unsound.

World came from Nirvana

I'm not sure how that statement is meant to work.

I see "the world" as being physical (rocks and stars and air and so on), also chemical and biological, also sensory (perception of the world), and social (the French word for "everyone" is tout le monde literally "all the world").

I see "nirvana" as being a state of mind, a state (or object) of perception, a state of wisdom (maybe ethics).

I suppose that a view of the world can come from an understanding of nirvana -- but I wouldn't say that the world comes from nirvana in e.g. the physical sense that one thing might come from (might be compounded from) another thing.

means world must have come out from state of Nirvana. But that is impossible

I think that Mahayana has doctrine a bit like this, about commonality between samsara and nirvana -- but, yes, I think that according to (in the context of the dctrine of) the Pali suttas that statement seems meaningless or confused.

can we at least conclude that cause or causes behind the world are without beginning?

Maybe. The doctrine doesn't say that dhammas are impermanent -- so perhaps they're "without beginning" -- and dhammas include physical laws.

I'm not sure that I'd call a dhamma a cause though -- I think of it as a description. As a scientist I wouldn't say ...

1. There's a dhamma which says the world must be like that
2. Therefore the world is like that

... (except as a "self-fulfilling prophecy" i.e. seeing the world in a certain way because you believe it to be so -- also called having a "view").

I'd say instead, something like ...

1. The world is such, it's as it is
2. Someone observes, "the world is such, as it is"
3. That observation or description is a dhamma
4. If the world is always like that, always behaves like that, always has the property which was described, then that dhamma is always true

In Buddhism, "the world" means subjective experience of the world (Umwelt).

The cause behind "the world" (aka "appearance") is called "Dependent Origination". Dependent Origination is eternal law.

Your statement that the world came from Nirvana is correct. This is called "Basic Nirvana" or "Primordial Nirvana".

In Mahayana we say that Primordial Nirvana and Final Nirvana are the same in theory but different in practice.

Primordial Nirvana is called "all-ground of various tendencies" (in the neutral sense). Or "innate ignorance" (in the negative sense). It has a natural tendency to evolve into confused mind which manifests as Samsara, and then this confused mind has tendency to further evolve into enlightened mind, which understands this entire process and its own ground, and accepts it as is - which is called Final Nirvana.

In the end, nothing really changes.