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

the monoid $M$ to the set $X$ and sends each element $f (- M$ to the corresponding map $m[f] : X --> X$ with $m[f ](x) = m(f,x)$ . The natural transformations between two such functors $m$ and $n$ (i.e. arrows in the category $M Sets$ ) are equivariant maps $h : X --> Y$ satisfying $ho m[f ] = n[f]o h$ for all $f (- M$ .

According to the properties of harmonic morphemes we have a monoid action $mX$ of $M$ on each chord $X (- U$ defined by virtue of $mX(f,x) = mX[f](x) = f (x)$ . Furthermore we consider two objects and of the category $M Sets$ together with two arrows between them, which allow us to study the internal logics of the category $M Sets$

Terminal object:: Recall that one-elemented sets are terminal objects of the category $Sets$ . Let ${*}$ denote an arbitrary fixed one-elemented set. Then there is a trivial action $m {*}$ of $M$ on ${*}$ with $m [f ](*) = * {*}$ for all $f (- M$ . If $m$ is any action of $M$ on a set $X$ the unique map $! : X --> {*} X$ is obviously equivariant and therefore $[m ] {*}$ is terminal in the category $SetsM$ .
Subobject classifyer:: Let $_O_ = {B (_ M |f o B (_ B for allf (- M }$ denote the set of all left ideals (cosieves) of $M$ and let $w : M × _O_ --> _O_$ denote the action defined by $w(f,B) = w[f](B) := {g (- M |go f (- B}$ . One easily verifies that $w(f,B)$ is always a left ideal. In particular we have $w[f](M ) = M$ and $w[f ](Ø) = Ø$ for all $f (- M$ . The left ideals play the role of generalized truth values, when we study characteristic functions with codomain $_O_$ .
Global truth values:: The equivariant maps: $Tt, _L : {*}--> _O_$ with $Tt(*) = M$ , $_L (*) = Ø$ are called the global truth values »true« and »false« respectively. The topos $M Sets$ with subobject classifier $_O_$ is called bivalent $,becauseTt$ and $_L$ are the only global elements of $_O_$ , i.e. the only equivariant maps $t : {*}--> _O_$ (Suppose $t(*) = B$ for a left ideal $B$ different from $Ø$ and $M$ , then we have $w(f,B) = M$ for each $f (- B$ and therefore $M = w[f]o t(*) /= t o m{*}(*) = B$ which violates the equivariance condition).

Remark 5 As soon as any of the maps $w[f] : _O_-- > _O_$ differs from the identity $1_O_$ the topos $M Sets$ is no longer well-pointed. This means that $w[f]$ is indistinguishable from the identity in terms of global elements, although it actually differs from the identity ( $w[f](M ) = M$ and $w[f](Ø ) = Ø$ implies that $w[f]o Tt = 1_O_ o Tt$ as well as $w[f ]o _L = 1_O_ o _L$ ). The category of sets is well-pointed but $M Sets$ is not. The internal logics of a morpheme is therefore different from classical logics. This becomes more clear when we study characteristic functions (see below).

To each action $m : M × X --> X$ and a subset $Y (_ X$ with $m(M ×Y ) (_ Y$ we can calculate the characteristic function $x : X --> _O_ Y$ of the equivariant set inclusion $Y (_ X$ by virtue of

x (x) := {f (- M |m(f,x) (- Y } (- _O_. Y

It is particularly interesting to study the characteristic functions $xY$ of chords as subsets of the full chord $X = T = {0,1,...,11}$ with respect to the monoid action of their perspective closure $M = ºA(X) = Int({X})$ (c.f. subsection 4.2).