Derivations

Linguists talk about lexical derivations when word stems through addition of derivative endings changes the word class.
Such transitions become very natural with application of the dimension model and word classes identified as have been done above.

Derivation in mathematics implies a stepwise decrease in degrees of a function as in the dimension model. In the enlarged context here the "derivations" go in both directions - and also within the same word class, for variation of sense: e.g. the verbs form - formalize. (See a Swedish example at end of this file.)
The derived word is "subject to" the special differentiations belonging to that word class which is created: tenses, cases, comparisons...

The general principle reminds of ionizations in chemistry when a molecule as for instance CH4 is ionized and the H-atoms are replaced by other elements.

The derivative endings seem to have a certain similarity with "determinants" in early ideograph writing, the special signs in hieroglyphic writing that were added for the meaning to get closer specified, for narrowing down the class of meanings.

The origins of lexical, derivative endings can be traced back to other verbs, nouns (also adjectives?) or rests of such morphemes. One Swedish example: the ending -lig transforms many noun stems to adjectives: form → formlig. This ending originates from a word lik = body, figure or shape.
Here we can remind of the 5 types of quality adjectives (RB) mentioned about adjectives before: 3 from verbs including participles, 2 from nouns.

The "noun disease", tendency to use substantivized verbs instead of verbs in formulations in our languages as Swedish and perhaps related ones, is e.g. apply - applying - the application of could lie inherent in the linguistic laws: the natural step of development from d-degree 4 to d-degree 3. This on the level of word classes, in outward direction. (Perhaps limited to some types of languages, as the inflecting ones?)
The many transitions from nouns to adjectives and "How-adverbs" is another: stone → stony, heaven → heavenly…

Number of derivative endings, used for the transitions between word classes (V-N-A...) in our related languages of today, are limited, which indicates that these derivations are part of a principle in a system, not only lexical.
They could in numbers and function be compared with the limited number of coenzymes in relation to protein enzymes in cell biology: coenzymes subdivided according to what they transfer: protons, electrons or small molecules, the "group-transporting" ones...

One reason for the limited number of these derivative endings could be that they at first and separately only represent the abstract structure of a word class, the steps between categories of words.

Calculating with both the primary and secondary dimension chains (the latter for differentiation within a word category) in a 2-dimensional coordinate system, we could get 5 x 5 derivative endings (in a first round?). 25 is also the number of coenzymes in the file referred to above.
Compare the 25 case affixes in Hungarian?

It seems to be an old linguistic question how lexical derivations relate to inflectional patterns (RB), to syntactic affixes, case endings etc.

Linguists have found many similarities between subcategories of case endings and lexical derivations (RB). Hence, it would be possible to regard these as results of corresponding principles on different levels of language.
The correspondences, which give the similarities, are here suggested to be dimensional differentiations according to the model here.

Cases are a differentiation within the word category of nouns.

- Cases -represent a secondary developed level, however with many syntactical implications, as the difference Subject - Object with impact on word order and with endings having adjectival sense as instrumental case etc.
Hence, cases have a relation to syntax, which the lexical derivative endings haven't.

A suspect figure:

- Both nouns as such and cases in the chain of differentiations within nouns are here identified as in d-degree steps 3 - 2, in different dimension chains related x → x'.
(The geometrical character becomes naturally of a more secondary derived type on the level of cases.)

- Compare too that in the super-chain of levels here the word categories as such are assumed identified in d-degree step 3-2 (in relation to Syntax in step 4-3 and Morphemes in step 2-1).

A third fact to mention here is that verbs, nouns - and adjectives often originate from the same morpheme (LB) or originate from verbs (in Indo-European and Semitic languages), e.g. grow - grass… This branching of morphemes could represent a third level, a deeper one in the gradient from phonemes towards whole sentences (in synthesizing direction).

Should these three things, morpheme branching, case endings and lexical derivations be regarded as three historical phases in development of IE and Semitic languages, creations of the same principle? And/or as referring to three different levels of language: the morpheme level - the word class level and syntax level in our scheme level 2 - 3 - 4 ? (Both the levels of word classes and of morphemes, 3 and 2, are more or less lexical.)

The geometrical poles out of step 3-2 are in our model elementary proposed as radial versus circular:

The lexical derivations may be regarded as the circular component, bridging over between word classes, while the branching of phonemes implies a radial component.
Or: from Syntax - in outward direction - from the whole sentence to phonemes: the radial component giving the word class differentiations. While, from the level of Morphemes - in inward direction - additions of morphemes (from reduced word class words) give the "circular" component, the bridges between word classes.
Note the two opposite gradients of the big level chain of linguistic analysis.

Compare also the two kinds of generality connected with 0- and 00-poles respectively, the deeper integrating one from a common centre, the 0-pole, and the superficial aggregating, surrounding one from outside, 00-pole, in terms of the dimension model.

The lexical character of derivations (something learned from outside) seems also emphasized in the statement (RB) that many such derivative endings are loans from other languages. (In accordance with RB's distinction between individual affixes and abstract lexical rules.)

Much in accordance with our model is the view (RB) on the relation cases - derivative endings as along 2 coordinate axes:

We should naturally regard each axis as representing a whole dimension chain and count on more than two.
   Affixes are seen as points on one axis with 1 coordinate, or in one of the plane quadrants with 2 coordinates, or in a space quadrant with 3 coordinates? The same affix can be used in more than one of the roles.
   Each coordinate axis can be moved along another one, which provides a basis for the many parallels between case forms and subcategories of "lexical derivations" that RB and others have pointed out.
    Perhaps the chain for differentiation of verbs should represent the 3rd axis with such aspects for instance as active-passive form, unlimited/limited action or in tense as completed or future / possible action. A relation between an axis for morphemes, one for word classes and this verb axis could relate the words drink (V) and drunk (Adj.) and drinkable (Adj.).