# Word for the relation between two different generalisations

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.

• That's quite abstract... can you give an example sentence that you'd like to use the word in? – psmears May 8 '11 at 7:11
• @psmears: "The hypercube is an X of the polygon, with an arbitrary dimensions instead of angles." There are probably non-mathematical examples as well. – user4727 May 8 '11 at 8:12
• I added the tag 'mathematics'; even though the request applies non-specifically, it is very relevant to mathematics and if any locution exists it would come out of that domain (or possibly software engineering). – Mitch May 8 '11 at 13:32

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.

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"

Perhaps you can say they are homologous?

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

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.

• I have never heard the phrase orthonormal generalisations, and Google heard it just once. Could you point me someplace where it is actually used? – user4727 May 8 '11 at 14:28
• Furthermore, I'd like to point to this relation without mentioning the common origin. – user4727 May 8 '11 at 14:28
• Hi Tim, no I can't. But mathematically each generalisation of a concept can be considered a vector in the concept space, then it just makes sense. alternatively relational algebra and category theory terminology could be used, but then we will be getting further away from the everyday english. – Arjang May 8 '11 at 14:44
• without mentioning the common origin? how about X and Y are (non-)orthogonal generalisations (with respective to one another of course) ? Now borrwoing from vectors analogy, the point of origin is immaterial. – Arjang May 8 '11 at 14:47
• The analogue is indeed interesting, but I'd like others to understand me as well :) – user4727 May 8 '11 at 14:59