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

Example 2 Suppose again that we are given a form semiotic $S$ with form set $F S$ denotator set $D S$ , over $M od@,@Mod$ , and with the above defined name form $N F$ with name value $“N ameF orm''$ . Given an identity morphism $~ Id : S --> S$ in $@ M od$ , we introduce two forms $N Fi : Id.Limit(Xi),i = 1,2$ , with diagrams $X1 : N F2 '--> S,X2 : N F1 '--> S$ and different name denotators $N F1 ~ N F2 : 0Z@N ameF orm(c : 0 '--> w),N F2 ~ NF1 : 0Z@N ameF orm(c : 0 '--> w)$ having the same name coordinate $c$ . Then the exchange of forms and corresponding names $N Fi$ defines an element of $GalMod@(S(N F1,N F2)/S)$ . Here, we have used the circular definition of forms by mutual reference.

Proposition 1 Every finite group is isomorphic to an automorphism group of type $Aut (S) G$ for a form semiotic $S$ over $M od@,@Mod$ , and $G < Aut(M od@,@Mod)$ .

It suffices in fact to consider any group $G < S(<UN ICODE >)$ of word permutations and to construct name denotators with name values that are permuted under $G$ over the form $“NameF orm''$ , which are their proper names.

Problem 1 The main problem of this Galois theory is to investigate the relation between conceptual constructions of forms and denotators, their associated automorphism groups and the existence problem of such concepts. This problem is evidently related to the underlying topoi and address categories. In view of this latter constraint, the question is also in how far concept constructions are a function of these topoi or else of a generic character, i.e., independent of the given topoi.

The problem is likewise manageable for regular extensions, i.e., for a set of forms an denotators the names and diagrams of which depend on an already existing form semiotic, whereas for circular, i.e., non-regular, extensions, very little is known. For example, as already mentioned above, one has some results concerning the existence of specific presheaves, see (Mazzola, 2002, G.2.2.1, Proposition 103). Concerning existence problems involving circularity, we suggest to review what Paul Finsler introduced in (Finsler, 1926) and what Peter Aczel calls Hyperset Theory (Aczel, 1988), (Barwise and Moss, 1991), together with associated techniques for the solution of set equations with circularity conditions as presented in (Pakkan and Akman, 1996).

References

ACZEL, PETER (1988). Non-well-founded Sets. Center for the Study of Language and Information, Lecture Notes Nr. 14, Stanford, CA, 1st edn.

BARWISE, JON and MOSS, LAWRENCE (1991). Hypersets. Mathematical Intelligencer, 13(4):31-41.

FINSLER, PAUL (1926). Über die Grundlegung der Mengenlehre. Erster Teil. Mathematische Zeitschrift, 25:683-713.

GROTHENDIECK, ALEXANDER (1985). Récoltes et Semailles. Université Univ. Sci. et Tech. Languedoc et CNRS, Montpellier, 1st edn