A full or correct sentence must include a subject and a predicate. The one apparent exception to this is a sentence in which the verb is in the imperative mood, which is indicated in English by the omission of the subject "you". Thus "Please be quiet!" or "Hurry up!" are both complete sentences with verbs in the imperative mood and an unstated subject ("you").
Incomplete sentences are not used in formal non-fiction prose, but are often used in ordinary conversation, text messaging, and in reported speech in fiction or non-fiction. In text messaging, many parts of a sentence may be omitted or abbreviated for brevity. Since, as a genre, it uses a limited number of conventional phrases, readers will normally understand the abbreviations used by their specific peer groups. "lol" as an abbreviation for "[I am] laughing out loud" is generally understood and accepted within the context of internet chat or text messaging but unacceptable in formal writing. The one exception to that would be if embedded with a direct or indirect quotation, for example:
She replied to his long, convoluted message with a single, three letter response: "LOL".