auxiliary
probably not; I doubt that
Expresses the speaker's negative conjecture or doubt about an action or state. Attaches to the dictionary form of verbs, or the stem of する (しまい).
He probably won't come anymore.
That probably isn't the case.
auxiliary
not intend to; have no intention of
Expresses the speaker's firm negative intention. Often used with the volitional form of verbs (e.g., するまい, 行くまい).
I will never do such a thing again.
auxiliary
Used in prohibitive or imperative expressions to strongly forbid an action. Often seen in formal or written contexts, such as rules or admonitions.
You must not make noise here.
He probably won't come again.
彼には何も言うまいと思った。
I decided not to say anything to him.
忘れるまいぞ。
Don't you forget!
She tried not to let her tears show.
I vowed never to drink again.
よう expresses the speaker's affirmative volition or conjecture ('will do/probably will'), the opposite of まい's negative volition or conjecture ('will not/probably not').
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).
Derived from the classical auxiliary まじ, which expressed negative conjecture or prohibition. The modern form まい is used primarily in written or formal speech.