In the USA, if someone writes that they are "professor" with no other specification, does this mean full professor, or can it also mean associate professor? In other words, is associated professor a subset of professor, or is it a fully distinct category?
In the US, Assistant and Associate Professors are also titled Prof. X when being referred to formally. This is unlike in Europe and parts of Asia, where "Professor" is a title that can be used only if they're a full professor. Others are simply referred to as Dr. X (or Mr. or Ms. X, if they don't hold a doctorate).
It should be pointed out that, at least in the US, "associate professors" are a partially overlapping set with the class of "tenured professor," as many associate professors have tenure, but have not yet been promoted to a full professorship. So, there really is "associate without tenure" and "associate with tenure."
However, in the US, as user7691 points out, the correct form of address for any professor, regardless of type, is "Professor X." I would even include adjunct and emeritus professors in this group. If you're looking at a faculty listing and see just "Professor" after a name (or in a separate field), however, it's likely that the individual in question holds a full professorship.
I thought about making this a comment as it only refers to Australia, and your question is about the United States, but it got a bit long.
The Australian Context:
- Professor is typically the top of the academic ranking hierarchy in the order: associate lecturer, lecturer, senior lecturer, associate professor, professor. Thus, professor means that someone is of the academic rank "Professor".
- Common general titles that cover all ranks include "lecturer", "academic", "researcher".
- In terms of titles, the basic rules are as follows (e.g., for Smith):
- If of professor rank, "Prof Smith"
- If of associate professor rank, "A/Prof Smith"
- If of associate lecturer, lecturer, or senior lecturer rank and the person has a doctoral qualification, "Dr Smith"
- If of associate lecturer, lecturer, or senior lecturer rank, and the peson does not have a doctoral qualification, "Mr Smith" or "Ms Smith"
Short answer: if somebody from the US writes about themselves as a Professor, then it means a tenured full professor.
Longer answer: If you see the title "Professor" in the official university publication (e.g., department website), or in somebody's bio, self-description, or email signature, then it means a tenured full professor (the third, and the top, step in the US academic hierarchy... Instructors and lecturers can hardly be considered academics as they are treated as staff for hire and dismissal).
If you see "Professor" in somebody else's email, especially students' emails, or in the university newsletter, or in some other source that was originated from a person that is not so familiar with the academic ranks, and the hoops one needs to jump through to get there, then it may mean a broader use of the term to indicate an instructor in a university. Essentially, all the adjectives (e.g., an Adjunct Assistant Professor... essentially nobody) are getting thrown out, with only the affiliation with academia remaining in this use of "professor" word. Such a liberty may qualify as an insult to a British/ANZ Professor where the meaning is way more specific. American academics are used to the confusions, though, and let the vague understanding of the title slip when used by un-initiated.
I was getting emails addressed "Professor" when I was in grad school... I would roll my eyes -- can't the source figure out a difference between a Ph.D. student and a regular faculty? They come on very different pages on the department website...