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

$R$ pointwise fixed. Conversely, given a subgroup $H < AutE(S)$ , there is a unique maximal subsemiotic $Sem(H)$ of $S$ over $E$ , which is left pointwise fixed under $H$ , much the same as there is a maximal subfield of a field which is left pointwise fixed under a given group of automorphisms of the given field. This defines a Galois correspondence Gal SubE(S)-----Sub(AutE (S)) Sem

between the set $SubE(S)$ of subsemiotics of $S$ over $E$ and the set $Sub(AutE(S))$ of subgroups of $AutE (S)$ .

Example 1 Suppose that we are given a form semiotic $S$ over $M od@,@Mod$ with form set $FS$ denotator set $DS$ with the above defined name form $N F$ with name value $“N ameF orm''$ . Suppose further that we are given a fixed diagram $X (- Dias(FS/M od@)$ . We may define two new forms $U ~ P N : Id.Limit(X),U'~ P N ': Id.Limit(X)$ as follows: The identifiers are the identities on the frames. The names are by definition two different zero-addressed denotators $' ' PN ~ PN : 0Z@N ameF orm(c),PN ~ P N : 0Z@N ameF orm(c)$ of the given name form $NF$ , represented by its name value. These denotators are their mutual name denotators. As to the coordinates $' c,c$ , we consider two situations:

In the first, we set $' c = c : 0Z --> Z<U NICODE >$ , which means that the only difference of names resides on the declaration that $' P N /= PN$ . Therefore, the form semiotic $' ' S(U,U ;PN, PN )$ defined as the extension of $S$ by the two new forms and the new names evidently has the automorphism $f$ over $S$ which exchanges forms and names, i.e., $' ' ' ' f(U ) = U ,f(U ) = U,f(P N) = PN ,f(P N ) = P N$ . This situation means that we have a purely conceptual automorphism in $' ' GalE(S(U,U ;PN, PN )/S)$ whithout touching any topos-theoretic >basic< data.

The second situation is the same for everything except that $' c /= c : 0Z-- > Z<UN ICODE >$ . In this case, suppose that the values $' c(0),c (0)$ differ from the given name values in $S$ . Then we consider the following construction of an automorphism in any category $C$ as follows: Suppose that we are given an automorphism $f$ of an object $A$ of $C$ . By definition, the automorphims $~ [f] : C--> C$ leaves all objects fixed. On the morphisms, we have four cases:

$Hom(A, A)$ is conjugated, i.e., $f '--> ff f-1$ .
For $X /= A$ , we have $Hom(X, A) ~--> Hom(X, A) : g '--> fo g$ , whereas
$~ -1 Hom(A, X) --> Hom(A, X) : g '--> g o f$ .
For $X, Y /= A$ , $Hom(X, Y )$ is left pointwise fixed.

We now apply this construction to $@ C = M od ,A = @Z <UN ICODE >$ , and $f = @Z <u >$ , and $u (- S(<UN ICODE >)$ , the symmetric group of words over $U NICODE$ . Using the transposition $f$ , this defines an element of the automorphism group $AutAut(Mod@,@Mod)(S(U,U ';PN, PN '))$ .