# What is it called when the definition of "<adjective> <thing>" does not imply that it is a special case of "<thing>"?

There are a couple of terminologies that I find mildly confusing:

• Using the definition of graph on wikipedia, a graph is defined as having a finite number of vertices and edges. It states that there is the concept of an infinite graph, but the term graph does not include them by default.
• Probably an even clearer is a non-deterministic Turing machine. Any time a Turing machine is defined, the definition is that of a deterministic Turing machine. A non-deterministic Turing machine is not actually a Turing machine under this definition.
• A Non-associative ring is not necessarily a ring or even non-associative.
• Traditionally, a plane is defined as being Euclidean. That is, if someone says in a traditional geometry textbook "let P be a plane", it is implied that it is Euclidean. When talking about non-Euclidean geometry, however, it can refer to either a euclidean or a hyperbolic plane, or even some other things, depending on context.
• The field with one element. It is neither a field nor does have one element. It is not even a set, necessarily.
• The most extreme example, at least to me, is the term non-standard as used in model theory. Though typically used when describing models, it can also be used to describe the elements of that model. In fact, for almost any mathematical concept, you can create a non-standard version of it by just applying that concept's definition within the model. For example, you can have nonstandard graphs, Turing machines, rings, planes, and fields. You can even have nonstandard versions of unique objects, like the real numbers or the free group with two generators. They will in general, however, not actual be an example of what term they are based on. A non-standard turing machine, for example, need not be a turing machine. Non-standard objects are perfectly well-defined objects, of course; they just aren't actually what their name seems to say they are.

Now, I am not arguing against such definitions. I know that once you get used to them, they can be quite convenient, and that they usually do not cause problems. After all, it would not make sense to say that something needs to fail associativity; we just sometimes do not mandate it. My question is what such a definition is called?

• Here is a reverse example: an <adjective> <thing> which is not <adjective>. An orthogonal projection matrix, which is a projection matrix but (except for trivial examples) is not orthogonal. Commented Feb 5, 2019 at 4:40
• These might be considered "abuses of terminology".
– Blue
Commented Feb 5, 2019 at 4:47
• It's not like these things only occur in mathematics. Consider "forged Picasso", "imitation crab", "fake fur", "honorary citizen", ...
– user856
Commented Feb 5, 2019 at 4:49
• A "generalized what-have-you" is an expansion of the definition of a what-have-you. One could see things like "infinite graph" as including an implicit "generalized".
– Blue
Commented Feb 5, 2019 at 4:54
• @Blue I guess you could say the adjective "generalized" is the "mother" of all these adjectives. Its also another example of where the adjective is not literal, since even though rado's graph could be said to be a generalized graph, it would not be correct to say that rado's graph is "generalized", whatever that would mean. Commented Feb 5, 2019 at 4:57