When a spade is not a spade

Typically, when one sticks an adjective in front of a noun, the resulting noun phrase refers to a subclass of the things that the bare noun refers to. For example, a red truck is a special type of truck.

There are certain standard exceptions, notably when the adjective serves a negating or broadening function. For example, a near success is not a success, and a would-be intellectual is not an intellectual.

Some of the most confusing terms in mathematics are those which violate the above principles. Three (admittedly rather arcane) examples that come to mind are:

• A quantum group is not a group
• A perverse sheaf is not a sheaf
• A Boolean-valued model is not a model

What are some other examples? I feel that there are probably many examples that I've gotten so used to that I no longer notice the "illogicality." I think it would be useful to compile a list of these so that people who teach math can be aware of them, and point out the possible confusion to students.

Note that examples involving adjectives such as "pseudo," "quasi," "almost," etc., don't really count in my book because these adjectives are widely understood to negate or partially negate the noun in question.

EDIT: Here is another example that occurred to me: A fractional ideal is not necessarily an ideal.

Solution 1:

Manifold-with-boundary is not (unless the boundary is empty) a manifold, a persistent source of confusion.

Also: "delta function." Sigh.

Others please feel free to add your contributions.

Solution 2:

A Hilbert-basis is not a basis.

Solution 3:

A rational function is typically not a function.