Word for the relation between two different generalisations

Question

I'm looking for a word for the relation between two concepts that are both different generalisations of the same concept. As an example, take the scalar product and cross product between vectors, from the product between scalars.

Answer 1

If I've understood the question right, you're looking for a word that can be used where

• You have a basic concept (e.g. multiplication of scalars)
• This concept is generalised in at least two ways A & B (e.g. A=the inner product / B=cross product of vectors)
• You want a word X that lets you describe this relationship - say "A is an X of B".

If that's the case, I don't know of a specific English word for that (it's a highly specific word, for a very abstract concept, so there may not be one), but if I had to describe this relationship repeatedly (say, when writing a mathematics paper) I'd probably use an analogy with family relationships - and perhaps use

The cross product is a cousin of the scalar product.

(or choose a different family member if that suits the purpose better). Of course, you'd have to say explicitly in the paper what you mean by using the term in this way.

Answer 2

You have three concepts X, Y, and Z, and Y is a generalization of X and also Z is a generalization of X. Then you want a simple way to describe the relation between Y and Z.

The only set pattern to say this is really:

Y and Z are two different generalizations of X.

For example, "the scalar and cross product are two different generalizations of the arithmetic product".

If you want to say "Y is (somethety something) Z" to relate Y and Z without reference to X, then there is also no set pattern (phrase or word), but you can say:

Y is a different generalization from Z

For example, "The hypercube is a different generalization (of the square) from a polygon".

If there is some more substantive relationship between Y and Z, say a concept that is a generalization of both Y and Z, or there is a specific direct relationship between Y and Z then you'd say just that (there is no set way to say it). (sibling, cousin, homologue all work but they are not set descriptions, and they imply something more than just "this generalization Y is another generalization in addition to generalization Z"

Answer 3

Perhaps you can say they are homologous?

homologous |hōˈmäləgəs; hə-| adjective having the same relation, relative position, or structure, in particular

[NOAD]

Answer 4

Conceptual Orthogonality is being used around ( thanks google ), orthogonal concepts can be understood to be completely different aspect than non-conceptually orthogonals, e.g. :

Size and Color are conceptually orthogonal to each other but Size and Height are not conceptually orthogonal.

If C is the main concept and X and Y generalise C in completely orthogonal manner with respect to each other, then X and Y are orthonormal generalisations of C, however if X and Y do not generalise C in compleltely different ways then it could be said that X and Y are non-orthonormal generalisations of C.

For example fractional calculus and varying the metric are orthonormal ways of generalising integration but complex integration and rieman integration are non-orthonormal generalisation of integration.