Transitive factorizations of free partially commutative monoids and Lie algebras
The free partially commutative monoid is defined by generators and relations as \(M(A,\theta)=\langle A\mid ab=ba,\;(a,b)\in\theta\rangle\), where \(A\) is an alphabet and \(\theta\subset A\times A\) is an ``independence relation''. The case of free partially commutative Lie algebras was studied in [\textit{G. Duchamp} and \textit{D. Krob}, Adv. Math. 95, No. 1, 92-126 (1992; Zbl 0763.17003)]. The authors consider factorizations of type \(M(A,\theta)=M(B,\theta_B)\cdot T\), where \(B\subset A\) is a subalphabet. The main result is a criterion to characterize the case when \(T\) is a free partially commutative submonoid. The bases of the associated Lie algebras are constructed as well. The case of the free partially commutative group is also considered.
scientific article; zbMATH DE number 1741703