The wine, cake

And can you say, if there is a cake on the table.
"Would you like a piece of the cake?" is that correct?
or do I ask
"Would you like a piece of cake?"

AnonymousI ask because I know you can't say "a bottle of the wine".

Yes, you can, in context:

I bought a bottle of the wine that was on sale.
I had a slice of the cake they gave me.

AnonymousAnd can you say, if there is a cake on the table."Would you like a piece of the cake?" is that correct?or do I ask"Would you like a piece of cake?"

You may say either; both host and guest know which cake is being spoken of. In your second sentence, 'cake' is uncountable.

Thank you very much, Mister. Micawber! I appreciate your response.
'Wine' can also be uncountable, right. So if a guest comes in and sees a bottle of wine on the table. If I understand you rightly. I can say:
"My friend, would you like a glass of wine?
Or
"My friend, would you like a glass of the wine?
Either is correct, yes?

And another question, I am sorry: if I treat cake as uncountable, do I say:
"I took a slice of cake. Then I ate the slice of cake."
Or do I say:
""I took a slice of cake. Then I ate the slice of THE cake."
I am talking about the same slice in the last 2 sentences in both examples.
Thanks so much.

Anonymous. I can say:"My friend, would you like a glass of wine?Or"My friend, would you like a glass of the wine?Either is correct, yes?

Yes, right.

Anonymousif I treat cake as uncountable, do I say:"I took a slice of cake. Then I ate the slice of cake."Or do I say:""I took a slice of cake. Then I ate the slice of THE cake."

Grammatically, you may say either, but mental consistency would have you say the first one.

Thank you, Mister. Micawber!