18,071 research outputs found
Generating operations of point algebras
AbstractIn this paper we define a class C of groupoids on Sn(=S × S × … × S) for arbitrary sets S and integers n ⩾ 2. For n a prime we give necessary and sufficient conditions in order for the binary operation (·) of (Sn, ·) in C to generate each of the binary operations (∗) for all (Sn, ∗) in C
Combinatorics of the free Baxter algebra
We study the free (associative, non-commutative) Baxter algebra on one
generator. The first explicit description of this object is due to
Ebrahimi-Fard and Guo. We provide an alternative description in terms of a
certain class of trees, which form a linear basis for this algebra. We use this
to treat other related cases, particularly that in which the Baxter map is
required to be quasi-idempotent, in a unified manner. Each case corresponds to
a different class of trees.
Our main focus is on the underlying combinatorics. In several cases, we
provide bijections between our various classes of trees and more familiar
combinatorial objects including certain Schroeder paths and Motzkin paths. We
calculate the dimensions of the homogeneous components of these algebras (with
respect to a bidegree related to the number of nodes and the number of angles
in the trees) and the corresponding generating series. An important feature is
that the combinatorics is captured by the idempotent case; the others are
obtained from this case by various binomial transforms. We also relate free
Baxter algebras to Loday's dendriform trialgebras and dialgebras. We show that
the free dendriform trialgebra (respectively, dialgebra) on one generator
embeds in the free Baxter algebra with a quasi-idempotent map (respectively,
with a quasi-idempotent map and an idempotent generator). This refines results
of Ebrahimi-Fard and Guo.Comment: Fixed errata about grading in the idempotent cas
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