# Free partially commutative structures

@article{Duchamp1993FreePC, title={Free partially commutative structures}, author={G{\'e}rard H. E. Duchamp and Daniel Krob}, journal={Journal of Algebra}, year={1993}, volume={156}, pages={318-361} }

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.

#### 51 Citations

The free partially commutative Lie algebra: Bases and ranks

- Mathematics
- 1992

Abstract In this paper, we study the free partially commutative Lie K -algebra L ( A , θ ) defined by a commutation relation θ on an alphabet A . Its behavior is very similar to that of the free Lie… Expand

On free p-superalgebras

- Mathematics
- 2000

In this article we consider several aspects of algebraic combinatorics and combinatorial algebra over fields of prime characteristics. P-super-Radford theorem gives the structure of the free… Expand

Transitive Hall sets

- Mathematics, Computer Science
- ArXiv
- 2005

The equivalence of the two properties of Lazard and Hall sets is proved and this allows to build new effective bases of free partially commutative Lie algebras. Expand

Gröbner-Shirshov Bases for Free Partially Commutative Lie Algebras

- Mathematics
- 2011

In this article, by using Composition-Diamond lemma for Lie algebras, we give a Gröbner-Shirshov basis for free partially commutative Lie algebra over a commutative ring with unit. As an application,… Expand

On Universal Equivalence of Partially Commutative Metabelian Lie Algebras

- Mathematics
- 2011

In this paper, we consider partially commutative metabelian Lie algebras whose defining graphs are cycles. We show that such algebras are universally equivalent iff the corresponding cycles have the… Expand

Universal equivalence of some countably generated partially commutative structures

- Mathematics
- 2017

We study universal theories of partially commutative Lie algebras, partially commutative metabelian Lie algebras, and partially commutative metabelian groups such that their defining graphs are trees… Expand

Universal equivalence of partially commutative metabelian Lie algebras

- Mathematics
- 2013

Abstract In this paper, we find a criterion for universal equivalence of partially commutative Lie algebras whose defining graphs are trees. Besides, we obtain bases for partially commutative… Expand

Parabolic and quasiparabolic subgroups of free partially commutative groups

- Mathematics
- 2007

Let Γ be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph Γ, which we call canonical parabolic… Expand

Universal Equivalence of Partially Commutative Lie Algebras

- Mathematics
- 2017

We study universal theories of partially commutative Lie algebras whose defining graphs are cycles and trees. Within each of the two above-mentioned classes of partially commutative Lie algebras,… Expand

Centralizers in partially commutative Lie algebras (in Russian)

- Mathematics
- 2012

In this paper, the complete description of centralizers of elements in partially commutative Lie algebras is obtained. The description is given explicitly in the terms of generators.