Invitations

There are many aspects of the Japanese language that utilize the concept of "indirectness." This becomes clear with the idea of invitations, or how you extend an offer to someone. If I want to ask someone to go see a movie, I would use the negative form of the verb, either with ません or ない, depending on formality. I cannot use the ます form to do this, because that would just sound like I'm asking a simple question.

リーさん、映画を見ませんか？
Lee-san, would you like to see a movie?

アヤト、図書館で勉強しませんか？
Ayato, would you like to study at the library?

ライト、買い物に行きませんか？
Laito, would you like to go shopping?

I believe that using the ない form is asking the exact same question, just with a different, more casual tone.

リーさん、そのレストランで晩ごはんを食べない？
Lee-san, do you wanna eat dinner at that restaurant?

アヤト、池の近いで泳がない？
Ayato, do you wanna go swimming at the nearby pond?

ライト、明日家にこない？
Laito, do you wanna come to my house tomorrow?

One last useful tip to note is that saying "no" in Japanese is rather uncommon. So if anyone asks me one of the above questions and I want to respond in the negative, I could say ちょっと. It literally means "a little" or "a bit", but in the context of these sentences, it would mean "not really" or "that's a little inconvenient for me."

アリス：十時に公園に行かない？
ジョン： ちょっと・・・

Alice: Wanna go to the park at ten?
John: It's a little... (inconvenient for me).