I know it's used for greetings in a restaurant or store. But what type of verb conjugation (ex: polite, plain, honorific, imperative, or something else) is it? Is it used with other verbs? Is it archaic Japanese or modern Japanese?
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It's used in modern Japanese, and I think it's frequently used with honorific language (though maybe not restricted to it as
It's polite imperative. It is derived from Edo-ben (as is all 丁寧語), but is still considered modern even though it isn't used outside 尊敬語 verbs.
Just to supplement the others, here is the scheme of ます：
未然形 (Irrealis form) - ませ (ましよ)
And also from nihongoresources further congujations of ます