# dendriform

## Definitions

• WordNet 3.6
• adj dendriform resembling a tree in form and branching structure "arborescent coral found off the coast of Bermuda","dendriform sponges"
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Webster's Revised Unabridged Dictionary
• a Dendriform Resembling in structure a tree or shrub; having a branching shape.
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Century Dictionary and Cyclopedia
• dendriform Resembling a tree; tree-like in form; arborescent; dendritic. Also dendritiform.
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Chambers's Twentieth Century Dictionary
• adj Dendriform den′dri-form having the appearance of a tree.
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## Etymology

Webster's Revised Unabridged Dictionary
Gr. de`ndron tree + -form,
Chambers's Twentieth Century Dictionary
Formed from Gr. dendron, a tree, and L. forma, form.

## Usage

### In literature:

It is not a pair of simple tubes, nor of dendriform tubes, but a closed network.
"Encyclopaedia Britannica, 11th Edition, Volume 2, Slice 3" by Various
Tracheae are developed which are dendriform and with spiral thickening of their lining.
"Encyclopaedia Britannica, 11th Edition, Volume 2, Slice 6" by Various
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### In science:

By this theorem, all dendriform dialgebra (resp. trialgebra) structures on V could be recovered from O-operators on the module (resp. on the algebra).
O-operators on associative algebras and dendriform algebras
Let (V, ≺, ≻, ·) be a dendriform trialgebra.
O-operators on associative algebras and dendriform algebras
Then it is straightforward to check that the dendriform trialgebra axioms of (V, ≺, ≻, · ) imply that (V, ·, L≻ , R≺ ) satis ﬁes all the axioms in Eq. (6) – (8) for a ( V, ∗)-bimodule k-algebra.
O-operators on associative algebras and dendriform algebras
Let (A, ≺, ≻, ·) be a dendriform trialgebra.
O-operators on associative algebras and dendriform algebras
Guo, Rota-Baxter algebras and dendriform algebras, J.
O-operators on associative algebras and dendriform algebras
Uchino, Quantum analogy of Poisson geometry, related dendriform algebras and Rota-Baxter operators, Lett.
O-operators on associative algebras and dendriform algebras
We generalize the well-known construction of dendriform dialgebras and trialgebras from Rota-Baxter algebras to a construction from O-operators.
O-operators on associative algebras and dendriform algebras
We then show that this construction from O-operators gives all dendriform dialgebras and trialgebras.
O-operators on associative algebras and dendriform algebras
Furthermore there are bijections between certain equivalence classes of invertible O-operators and certain equivalence classes of dendriform dialgebras and trialgebras.
O-operators on associative algebras and dendriform algebras
This paper shows that there is a close tie between two seemingly unrelated objects, namely Ooperators and dendriform dialgebras and trialgebras, generalizing and strengthening a previously established connection from Rota-Baxter algebras to dendriform algebras [1, 2, 13].
O-operators on associative algebras and dendriform algebras
On the other hand, with motivation from periodicity of algebraic K -theory and operads, dendriform dialgebras were introduced by Loday in the 1990s.
O-operators on associative algebras and dendriform algebras
P y = xP(y), x ≻P y = P( x)y, ∀ x, y ∈ R, deﬁne a dendriform dialgebra (R, ≺P , ≻P ).
O-operators on associative algebras and dendriform algebras
This deﬁnes a functor from the category of Rota-Baxter algeb ras of weight 0 to the category of dendriform dialgebras.
O-operators on associative algebras and dendriform algebras
These studies further suggested that there should be a close relationship between Rota-Baxter algebras and dendriform dialgebras.
O-operators on associative algebras and dendriform algebras
Then it is natural to ask whether every dendriform dialgebra and trialgebra could be derived from a Rota-Baxter algebra by a construction like Eq. (3).
O-operators on associative algebras and dendriform algebras
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