"I don't have a son."
Translation:У мене немає сина.
Both ways are possible. «Я не маю сина» should definitely be accepted.
In Ukrainian as spoken in Ukraine, «у мене» means not just 'in my place' (although this is a possible meaning, too, and your understanding is also correct), but it's also a generic-purpose preposition to express possessor. Basically, «у мене є/немає X» means the same thing as «я маю/не маю X».
Ah! Now I understand! This "є" at the end of "нема" along with the "у" at the beginning is actually a Russification of the Ukrainian language! Aга!! Тепер зрозуміла! Час найвищий щоб вернутися назад до 'старої мови' щоб захищати нашу мову і душу! Треба нам затримати 'чисту Українську мову' з Українськими словами, і залишити 'позичині' слова з інших країн що би іти справжно на вперід!
It's not russification, this construction has existed since time immemorial. Similar construction exists in Latin, for example, mihi est fīlius 'у мене є син' which basically follows a similar patten. Also, Russian form is «нет», not «немає». «Немає» is an Ukrainian-only form, not found in neighbouring languages.
I could say you construction is a Polonisation. Because Polish uses 'mieć' «мати», Russian uses «у» + X. Ukrainian people lived between Polish and Russians, so they have both constructions. Neither is more 'Ukrainian'.
(N.B. It's not actually Polonisation, just like «у» is not Russification. «У» is a shared common element of Ukrainian and Russian, because they are both East Slavic languages. «Мати» and mieć is a shared element of Ukrainian and Polish, because they are both Slavic languages — Russian has such a verb too, име́ть, but it has fallen out of use except some bookish texts.)
«Немає» was used both by Ukrainian classic writers and by modern Ukrainian writers, like I've shown you in a different thread. If Shevchenko has no authority for you, then who has? You're eschewing the most famous Ukrainian classics as 'Russified' to promote your own dialect (which is probably Polonised, but you conveniently don't notice this fact). But why should we follow you and not follow the Shevchenko? No, really, can you name just one reason why your language is better than the language of Ukrainian classics?
«Я не мене син» is not a possible sentence in Ukrainian, it's something like "I not me son".
There are two ways to express 'not having'. First, you can use the verb ма́ти 'to have', like in English or Polish:
- Я не ма́ю си́на. 'I don't have a son.' (Literally, 'I not have son.')
Second, you can use the expression «немає» 'there is no', and use «у» to indicate possessor, like in Russian:
- У мене́ нема́є си́на. 'I don't have a son.' (Literally, 'At my [place/possession], there-is-no son.')
Unlike «є» 'there is', «нема́є» 'there is no' is used with the genitive forms of the noun, so that's why you need to use «си́на» and not «син»:
- У мене́ нема́є си́на. 'I don't have a son.' (Literally, 'At my [place/possession], there-is-no son.') — нема́є requires a genitive form, сина
- У мене́ є син. 'I have a son.' (Literally, 'At my [place/possession], there-is son.') — є requires a nominative form, син