There are some Japanese tokenizers such as Kuromoji. When using the tokenizer on the expression しましょう, it gave the following analysis:
Surface form Part-of-Speech Base form Reading Pronunciation し 動詞,自立,*,* する シ ましょ 助動詞,*,*,* ます マショ う 助動詞,*,*,* う ウ
Somebody noted that it doesn't seem to make sense to separate ましょ and う, and he believes that grammatically -mashō should belong together.
In the Wiktionary entry for う, it is noted that う is used to make the volitional form of verbs, but "the volitional form of Group II (ichidan) and Group III (irregular) verbs are formed with よう (-yō) instead of う (-u)."
Interestingly, if I change "しましょう" to "しましよう" (note the way the よ is written), the tokenizer gives the below results instead:
Surface form Part-of-Speech Base form Reading Pronunciation し 動詞,自立,*,* する シ まし 助動詞,*,*,* ます マシ よう 名詞,非自立,助動詞語幹,* よう ヨウ
Could it be the case that する is just a special word, and due to the way "ょ" is written in this word, the tokenizer just failed to recognize "よう" as an entity? So the correct grammatical interpretation should still be the later table of results, right?
EDIT: I guess this (breaking the expression into three parts (しーまし(ます)ーよう), i.e. (do-polite-volitional)) might just be some formal grammar analysis thing. Possibly few people think in this way (i.e. breaking this expression into three parts) in their daily lives. So maybe many people on this SE would find it weird.