ませんか

(Mase n ka)

won't you...? / would you like to...? (invitation).

Connection rules

＋Verb ます-stem + ませんか (the same stem used for the ます form).
＋Functionally an invitation, not a literal yes/no question despite the negative shape.
＋Compare with 〜ましょうか (shall we) — ましょうか assumes the other party is amenable, while ませんか leaves more room to decline.

一緒いっしょに映画えいがを見みませんか。

Issho ni eiga o mi mase n ka.

Would you like to watch a movie together?

「won't you...? / would you like to...? (invitation)」

明日あした、一緒いっしょに食事しょくじしませんか。

Ashita, issho ni shokuji shi mase n ka.

Would you like to have a meal together tomorrow?

「won't you...? / would you like to...? (invitation)」