This is an auxiliary which expresses the negative volition or conjecture of the speaker. It is basically a written form, used in formal writing.
For Gr. 2 verbs, Ｖmasu is occasionally used instead of Ｖinf・nonpast. For irregular verbs, すまい and こまい are sometimes used instead of するまいand くるまい, respectively. See examples #6088, #6089, and #6090.
まい does not have a past form. The past tense is expressed by other verbs. See examples #6091 and #6092.
The volition of someone other than the speaker can be expressed by まい, as in example #6093. Note that と思う is used to express the volition of someone other than the speaker. Thus, example #6094 expresses the conjecture of the speaker rather than the volition of Keiko.
Ｖinf・nonpastまいとする expresses the idea of "try not to Ｖ." See example #6095. When this structure is used in a subordinate clause, する is often omitted, as in example #6096.
ＶvolitionalかＶinf・nonpastまいか expresses the idea "whether (I) will Ｖ or not," as in example #6098.
Ｖnonpastまい 1. 行くまい (〜will not go; probably will not go) 2. 教えるまい (〜will not teach; probably will not teach)
Adj(i)stemくはあるまい 1. 高くはあるまい (〜is probably not expensive)
Adj(na)stem/Nounであるまい 1. 静かではあるまい (〜is probably not quiet) 2. 先生ではあるまい (〜is probably not a teacher)
I`m not sure if my example #8089 is really this -mai.
Is this similar to あるわけない？
まぬかれる is a dictonary form. It is also pronounced as まぬがれる. 免れる means 逃れる. まぬがれ + まい (neg) >>> You will not be able to escape from being punished this time.
in #4703 what does まぬがれまい mean?
As for the first example, we don't say 死ぬまい for the third person. We'd say 病気で死ぬことはあるまいと思うが、…。 When you talk about yourself, you'd say この病気で死ぬまい、絶対生きてやる。
This usage is now generally replaced by ないだろう
Commonly used with あるまい
Hmm, my example might be 'mai-3' : expression of a negative that is natural or appropriate.