"Monday is a day."
Translation:Η Δευτέρα είναι μία μέρα.
It's not incorrect. I could add it as an alternative (could, because this one can't be found in the incub for some reason), but I think it sounds just a bit odd. That's because when someone says "Είναι μέρα" in Greek, they usually mean that it's day in general (as in not night. A general case, which is why the article is of course ommited), and not a day of the week. Personally, I think that "H Δευτέρα είναι μέρα της εβδομάδας" would sound more natural, but that might be just me. ^.^
Not really. Definite articles in greek can't be ommited for days of the week or months of the year, except for some cases.
In my opinion,(as a native speaker), you can omit the article when the day of the week or the month of the year is an object in a sentence and not a subject. For example:
Monday is my favorite day of the week - Η Δευτέρα είναι η αγαπημένη μου μέρα της εβδομάδας. (said no one ever.) subject, so the article can't be omitted.
January is the first month of the year. - Ο Ιανουάριος είναι ο πρώτος μήνας του έτους. subject, so the article can't be omitted.
It was Monday - Ήταν Δευτέρα. not a subject, so it's omitted.
It's already June. - Είναι ήδη Ιούνιος. not a subject, so it's omitted.
Also, there are cases that you can choose whether to omit it or not. Like
I'll call you on Monday morning - Θα σου τηλεφωνήσω (την) Δευτέρα (το) πρωί.
I hope this helped you a bit. ^.^