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From what I remember reading, a sentence in Japanese must either end with an adjective, a copula or a verb, if we exclude sentence ending particles and inflections. This being the case, I'm under the belief that "元気" by itself does not constitute a grammatically correct sentence in modern Japanese. Is this correct?
Furthermore, I'd like to know how a Japanese sentence is rigorously defined and what components are necessary.
Yes, I believe it's a Japanese sentence (sometimes called a minor sentence or a sentence word).
Looks like many English speakers were taught at school that a sentence must contain a predicate (a verb or a copula). However, as a native Japanese speaker, I have never heard such a rule from Japanese teachers in schools in Japan. Things like 元気。, はい。, おはよう。 or わ！ are always perfectly grammatical Japanese sentences to me. Simply, any valid expressions delimited by periods (or question/exclamation marks) are grammatically sentences. Some consider these "incomplete" sentences, but they are not even incomplete to me; there is no "full version" of こんにちは, for example.
If you look at the Wikipedia article about sentence, you can see many linguists think a sentence does not have to contain a predicate. Something like "Mary!" or "Wow!" are sentences to them.
Even from a grammatical point of view, 元気 is a perfectly fine Japanese sentence.
From what I remember reading, a sentence in Japanese must either end with an adjective, a copula or a verb, if we exclude sentence ending particles and inflections.
This is more or less accurate. Here, though, we need to see that 元気 falls into the category of "ending with a copula", but with the copula itself omitted. "元気。" or "元気？" in its full version can be
- 元気だ。
- 元気です。
- 元気ですか？
- 元気です？ (Yes people generally add か but grammatically it's not NECESSARY)
It just happens that だ can be omitted, and です, although not omitted because it's respectful language (omitting is considered disrespectful or casual), there is nothing grammatical that stops these copulas from being omitted.
Therefore, "元気", whether a statement or a question, is a completely grammatically correct Japanese sentence. If you have time, I suggest reading this chapter of Tae Kim's Guide to Japanese Grammar which does explain on this topic.
