Why "being" was not eliminated?

Question

1- I come across with below sentence written by a professor from the US in a paper:

All amorphous material are actually in states which are configurationally frozen, a particular configuration being specified by its partial distribution.

it seems, here, a reduced relative clause was used, right?
Why "being" was not eliminated?
I mean that the sentence should not be like:

.... a particular configuration specified by its partial distribution.

I mean I expect something like this we should have:

The car, which was purchased in Seattle, was a vintage Mustang.
The car purchased in Seattle was a vintage Mustang.

2- Additionally, I found below sentence in a grammar book:

The fruit was expensive, being imported. (The fruit was expensive because it was imported)

They used participle clause, here. To the best of my knowledge, we can rewrite above sentence like below,

The imported fruit was expensive.

which is thanks to leaving one word after reduction. Now, what if we have something like below,

The fruit was expensive because it was imported + (from a few countries by some cartels OR a clause)

to exercise participle clause for above sentence, which one is better, and why?

1- The fruit was expensive, imported from a few countries by some cartels.
2- The fruit was expensive, being imported from a few countries by some cartels.
3- Imported from a few countries by some cartels, the fruit was expensive.
4- Being imported from a few countries by some cartels, the fruit was expensive.

Answer 1

A relative clause will meet three requirements:

1. It will contain a subject and a verb.
2. It will begin with a relative pronoun (who, whom, whose, that, or which) or a relative adverb (when, where, or why).
3. It will function as an adjective, answering the questions What kind? How many? or Which one?

The relative clause will follow one of these two patterns:

• Relative pronoun or adverb + subject + verb
• Relative pronoun as subject + verb

Example 1)

All amorphous materials are actually in states which are configurationally frozen, a particular configuration being specified by its partial distribution.

The part in italic is not a reduced relative clause, because adding "which" or "that" before it does not make sense and does not make it a "normal" relative clause.

Example 2)

2- The fruit was expensive, being imported from a few countries by some cartels.

4- Being imported from a few countries by some cartels, the fruit was expensive.

The sentences 2 and 4 sound fine. In both cases, "being" means "because it was."

1- The fruit was expensive, imported from a few countries by some cartels.

Above, if you skip "being," the meaning of "because it was" is lost and the sentence just shows two independent facts: 1) The fruit was expensive and 2) imported.

3- Imported from a few countries by some cartels, the fruit was expensive.

I believe in the above case, "imported" alone might work as "being imported" in the sense "Because it was imported."

Answer 2

All amorphous material are actually in states which are configurationally frozen, a particular configuration being specified by its partial distribution.

In this sentence, being is necessary because the participial clause is in passive voice. This is what it would look like in active voice, without the being:

All amorphous material are actually in states which are configurationally frozen, its partial distribution specifying a particular configuration.

Note that this sentence is inconsistent in its use of number: material is singular, are is plural, states is plural, and its is singular. It should probably be like this:

All amorphous materials are actually in states which are configurationally frozen, their partial distribution specifying a particular configuration.

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The fruit was expensive, being imported

This passage from the BBC learning English web site explains the usage of being in this sentence:

being in participle clauses
We can use an adverbial participle clause to express reason or cause as an alternative to a because/since/as clause. Using a participle clause in this way is more characteristic of written English or a literary style, rather than spoken colloquial English.

In your four alternative sentences, 2 and 4 correctly and clearly express causality, whereas 1 and 3 merely hint at causality. Generally, you should place a participial clause nearest to the noun that it relates to. In such a short sentence, the difference is trivial, but Fruit is closest to the start of the sentence, so option 4 is the marginally better version.

4- Being imported from a few countries by some cartels, the fruit was expensive.