Free partially commutative structures

Abstract In this paper, we study algebraic structures defined by a presentation of the form 〈 A; ab = ba for (a, b) ∈ I 〉. We show that these relators are the only one that can be interpreted as both monoid and Lie relators. We also study the free partially commutative monoids, Lie algebras, and groups: in all of these cases, we show how to obtain decomposition results for these structures into the corresponding free structures.