Short answer
If the genitive is a Modifier within a larger noun phrase, then it will have no effect on whether we use an article or not. This will just depend on whether the head noun in the larger noun phrase is plural. Modifiers in noun phrases do not have their own Determiners.
However, if the genitive noun phrase has the function of Determiner within the larger phrase, then it may need its own article. This will depend on whether the head noun in the genitive NP is singular. If it is it will require a mandatory Determiner of its own (possibly an article such as a).
Full answer
- [a] [dogs' home].
- [dogs' homes]
- [a dog]'s [home]
- [a dog]'s [homes]
- [the dogs]' [home]
- [dogs]' [homes]
Just like clauses have two main parts, a Subject and a Predicate, noun phrases have two parts too. They are made up of a Determiner and a Head. In a pen the word a is the Determiner, and pen is the Head. If we add an adjective to the noun phrase it will go inside the Head where it modifies the head noun. So in an expensive pen, an is the Determiner and expensive pen is the Head.
Examples 1 & 2
In example (1) above we see the simple noun phrase "a dogs' home". It might help if we give an example sentence:
- My friend couldn't look after my dog this holidays, so I had to put her in a dogs' home.
In this noun phrase the Determiner is a and the Head is dogs' home. Inside the Head we can see home being modified by the word dogs'. Often we see adjectives doing the job of modifying the head noun, but we can also see nouns doing it too, for example: a gas cooker. In this instance we can see a genitively inflected noun (technically a nominal) dogs' doing this job. Here the word dogs' isn't telling you the owner of the home, it is telling you what type of home it is. Notice that because the head noun, home is singular, the noun phrase must have a Determiner, in this case the word a.
Example (2) is the plural form of (1) where there is more than one home being considered. It is because the head noun homes is plural that this is a plural noun phrase. Because homes is plural we don't need a Determiner at all. Here's an example sentence:
- This holidays I kenneled my two dogs at different dogs' homes.
So this noun phrase has a Head, but no Determiner.
Examples 3 & 4
Example (3) doesn't tell us what type of home we are talking about, but it tells us the owner of the home. Here's an example sentence:
- A dog's home is its castle.
In this large noun phrase the Head is home. The Determiner is not the word a but the smaller noun phrase a dog's, which consists of the noun phrase a dog along with the genitive particle/clitic/inflection 's. Notice that inside this smaller noun phrase the head noun dog is singular and so this smaller noun phrase requires its own Determiner, the word a.
The plural version of this noun phrase is example (4). Here we have more than one home:
- A dog's homes are its kennel and its master's living room.
But we still only have one dog. The Determiner here is the singular noun phrase a dog's. Because there is only one dog, we still need the Determiner inside this smaller noun phrase, the word a. Of course we don't need a Determiner for the larger noun phrase because the word homes is plural. We can easily have sentences like:
- Homes are very important to dogs.
But if we want to show whose homes we are talking about then the Determiner is important here for showing the possessor.
Examples 5 & 6
In (5) we have the Determiner the dogs', where the plural noun dogs' indicates that this single home is owned by two or more dogs together:
- The dogs' home is an old shed in an abandoned field. They like it very much though.
Here the head noun home is singular and so we must have a Determiner when this word is used with this meaning. Again as with example (4) when homes is plural we do not need a Determiner, but if we want to show the possessor of the homes in question then we will need to use a genitive noun phrase as a Determiner:
- Dogs' homes should be kept warm in the winter.
Notice that because the noun phrase in the Determiner is plural, it doesn't need a Determiner of its own. However, if we want to show the possessor we will need to use a genitive noun phrase as a Determiner. The head noun in the noun phrase inside the Determiner, dogs, is plural. For this reason, it does not require its own Determiner.
Conclusion
If the genitive is a modifier within a larger noun phrase, then it will have no effect on whether we use an article or not. This will just depend on whether the head noun in the noun phrase is plural. Modifiers in noun phrases do not have their own Determiners.
However, if the genitive noun phrase has the function of Determiner in a larger phrase, then it may need its own article. This will depend on whether the noun in the genitive is singular or not. If it is plural it will not need a Determiner of its own, if it is singular it will.
- A friend's book (article needed within Determiner because of singular friend)
- A friend's books (article needed within Determiner because of singular friend)
- People's need to be appreciated (no article needed within Determiner because of plural people)
- People's needs (no article needed within Determiner because of plural people)
- A ladies' wristwatch (the Head is singular phrase ladies wristwatch, article required)
- Ladies wristwatches (the Head is plural phrase ladies wristwatches, no article required)