auxiliary verb
indicates invitation; shall we?; why don't we?
Used to gently invite or suggest an action to the listener. Often softened with か (e.g., 行こうか).
Shall we head out soon?
He has probably already gone home.
一緒に映画を見よう。
Let's watch a movie together.
Why don't we have some tea?
まい expresses negative volition or negative conjecture (行くまい 'I won't go' / 'probably won't'), the opposite of the positive intention or guess expressed by う (行こう).
よう is the parallel volitional/conjecture auxiliary used with ichidan and irregular verbs (食べよう, しよう), while う is used with godan verbs (書こう) — learners must pick the right one based on conjugation class, not meaning.
Derived from the classical auxiliary verb む (mu), which expressed volition or conjecture. Over time, む shifted to う in modern Japanese, becoming the standard volitional ending for godan verbs.