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
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
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.