Determinants and Möbius functions in trace monoids
Let \(\Sigma\) be a finite alphabet, let \(I\subset\Sigma\times\Sigma\) be a symmetric and irreflexive relation on \(\Sigma\), and let \(R\) be the smallest congruence that extends the set of equations \(\{ab=ba;\;(a,b)\in I\}\). The quotient \(\Sigma^*/R\) is called a trace monoid. The Möbius function of \(\Sigma^*/R\) is defined by (\(\varepsilon\) being the empty word) \[ \begin{cases}\mu(\varepsilon)=1\\ \mu(a_1\cdots a_n)=(-1)^n\quad &\text{if }a_i\in\Sigma,\text{ all the }a_i\text{'s are different and }\forall i,j\;i\neq j\Rightarrow(a_i,a_j)\in I\\ \mu(t)=0\quad &\text{otherwise}.\end{cases} \] The authors prove in particular that, if the independence relation of a trace monoid \(M\) admits a transitive orientation, then any unambigous lifting of the Möbius function of \(M\) is the determinant of a matrix defined by the smallest acceptor of the corresponding cross-section.
scientific article; zbMATH DE number 1321849
scientific article