Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

If $T(F) = Power$ , then $D(F)$ is a diagram with one vertex form $G$ , $D(F )$ has no arrows, we have $D(F )(G) = space(G)$ , and $space(G) frame(F ) = _O_$ .

If $T(F ) = Limit$ , then for all vertexes $Gi$ of $D(F )$ , we have $D(F )(Gi) = space(Gi)$ , and $frame(F ) = lim D(F )$ .

If $T(F) = Colimit$ , then for all vertexes $Gi$ of $D(F)$ , we have $D(F )(Gi) = space(Gi)$ , and $frame(F ) = colim D(F )$ .

Let us give some comments on this definition with respect to the previous approaches as documented in (Mazzola, 2002).

Remark 1 In reasonable programming contexts, it is assumed that the names of forms are keys, i.e., that the application $F N$ is injective. In this case, a diagram can also be considered with vertexes being form names instead of forms. The denotator names are also denotators now, instead of being elementary character strings. This approach is a substantial enrichment compared to the usually string-oriented naming technique, as described, for example, in (Schewe, 2000). This generalization enables us to work with global and more complex name spaces. In particular, this enables us to assume that like form names, denotator names are also keys, i.e., that $DN$ is injective. And it opens the path to more structured name space concepts by use of forms which are especially designed for name management.

For example, if $@ E = M od$ and $@ R = @M od$ , we may consider the monoid $Z$ -algebra $Z<U NICODE >$ over the set $U NICODE$ of Unicode symbols and its represented presheaf $@Z <UN ICODE >$ address object. Then we have this form $N F ~ fn : Id.Simple(Dg)$ , where $Dg = @Z <UN ICODE >$ , $Id$ is the identity on $Dg$ , and $fn ~ fn : 0@F N (C)$ , the coordinate $'' C : 0 --> Z <U N ICODE > : 0 '--> “N ameF orm$ representing the corresponding natural transformation $@0 --> @Z <UN ICODE >$ . Observe that the denotator $fn$ is its proper name denotator. To this initial naming tool, we may add any name denotator $n ~ n : 0@N F(C : 0 '--> Anyname)$ with $n /= fn$ , and $Anyname (- Z <U N ICODE >$ , for example $'' Anyname = “3.V iolin+ 4.Piano$ . On this construction level, we have a single form, and a number of denotators for this form, each of them being zero-addressed, and its own name denotator, i.e., it essentially identifies to its coordinate value $Anyname$ .

Remark 2 In former setups for form semiotics, we had included the type “Synonymy” in order to allow plain changes of form names. This feature is easily realized by use of type $Limit$ , and a one-point diagram, similarly to the diagram used for $Power$ type. Therefore one is dispensed from this type.

Remark 3 There is a subtlety in the diagram definition which one should observe. In a sloppy language it is possible to have diagrams of objects in a category such that one and the same object appears in more than a single vertex. This is correct since diagram schemes are charged with the “indexing” job. Here, we do not have arbitrary diagram schemes, their vertexes must stem from the form set $F$ . This is not an impoverishment of mathematical expressivity, it is just the strict duty to use nothing except what is given in the form semiotic in order to comprise all features without being forced to invent new names and symbols on the flight.