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The color of house that I built is red.

This sentence can be written in following way.

The color of house I built is red.

I can omit that, because that is indicating the house, and it is the object of the verb built. A relative pronoun can be omitted if it indicates the object. But if it indicates the subject, it can not be omitted. Am I right?

Let's see the following sentence.

Apparel engineers are responsible for establishing and monitoring processes essential to maintain product consistency on time production and fair treatment of workers.

In this sentence, I think that has been omitted before essential. I can write this sentence in the following way.

Apparel engineers are responsible for establishing and monitoring processes that are essential to maintain product consistency on time production and fair treatment of workers.

Is it correct? Here that has been used to indicate to processes. In that are essential to maintain ... the relative pronoun that is a subject, not object. Please tell me why that has been omitted before the word essential? Please give proper explanation with some effective examples.

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  • The problem is that the first noun phrase in both sentences is already ungrammatical -- house should have a definite article. And the title is also ungrammatical. As for deleting that, any relative pronoun may be deleted, optionally, when it is not the subject of the relative clause it introduces. Since the subject is I, that may be deleted, and so may which. Indeed, if it's deleted, it's not clear whether it was sposta be which or that -- another speaker's choice foreclosed by deletion.
    – John Lawler
    Commented Apr 6, 2015 at 15:53
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    That's because type is not the same kind of noun as color. That's why naming "parts of speech" is pointless -- not all adjectives (or nouns, or verbs) follow the same rules as others. Pretty much, every lexical item has its own history and idioms and habits and affordances. There are practically no useful grammatical rules that depend on "Noun" or "Adjective" alone. Color nominals and classificational nominals are a good example of this.
    – John Lawler
    Commented Apr 6, 2015 at 16:09
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    ... There are many examples of 'the colour of car' on the internet, and I'm quite familiar with the usage. eg Do you know that the colour of car you drive can reveal a great deal about your personality? This doesn't jar with me at all. Perhaps it's become more accepted in the UK than in the US.
    – Edwin Ashworth
    Commented Apr 6, 2015 at 16:59
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    @JohnLawler Yes, I must agree with Edwin. If it is wrong or non-idiomatic then I have been making the error all my life. The colour of car I prefer, never seems to be available in the nearly-new market.
    – WS2
    Commented Apr 6, 2015 at 17:45
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    In color of car, car is being used to refer to the general category or concept of cars, so no article is used. In color of the car, it's referring to a specific car, so we use an article.
    – Barmar
    Commented Apr 6, 2015 at 18:55

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You are correct. The phrases presented are relative clauses. Typical relative clauses are introduced with a relative pronoun (who, that, etc.), but this is not necessary. A relative clause that excludes the pronoun has an elliptical relative pronoun.

The flavor I love is vanilla.

The elliptical relative pronoun is "that" to create this sentence:

The flavor that I love is vanilla.

In your longer example, you are correct in noticing that the adjective clause describing the type of processes possesses an elliptical relative pronoun.

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