Determinants and Möbius functions in trace monoids
type: Item
instance of: scholarly article
title: Determinants and Möbius functions in trace monoids
zbMATH Open document ID: 0929.20043
DOI: 10.1016/S0012-365X(98)00211-8
author: Christian Choffrut
author: Massimiliano Goldwurm
published in: Discrete Mathematics
publication date: 2000-01-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20M35
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 68Q45
zbMATH DE Number: 1321849
zbMATH Keywords: rational formal power series
zbMATH Keywords: finite alphabets
zbMATH Keywords: congruences
zbMATH Keywords: trace monoids
zbMATH Keywords: Möbius functions
zbMATH Keywords: determinants
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
cites work: Q3859267
cites work: Q3793747
cites work: Combinatorial problems of commutation and rearrangements
cites work: Q3489462
cites work: Combinatorics on traces
cites work: Möbius functions and confluent semi-commutations
cites work: Automata in general algebras
cites work: Q3853827
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description: scientific article; zbMATH DE number 1321849