A remark on the representation of trace monoids (Q584429)

The trace monoids are the quotients of free monoids by the congruence generated by some partial commutations of the letters [cf. \textit{P. Cartier} and \textit{D. Foata}, Problèmes combinatoires de commutation et réarrangements (Lect. Notes Math. 85, 1969; Zbl 0186.30101) and \textit{G. Lallement}, Semigroups and combinatorial Applications (1979; Zbl 0421.20025); see also \textit{IJ. J. Aalbersberg} and \textit{G. Rozenberg}, Theor. Comput. Sci. 60, 1-82 (1988; Zbl 0652.68017) and \textit{A. Mazurkiewicz} in Proc. MFCS'84, Lect. Notes Comput. Sci. 176, 115-133 (1984; Zbl 0577.68061)]. The main result is the following Theorem: A trace monoid can be faithfully represented by \(2\times 2\)-matrices with entries in \(N\) if and only if it is a direct product of a free commutative monoid with a free product of free commutative monoids.