dendriform


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dendriform

 [den´drĭ-form]
tree-shaped.
Miller-Keane Encyclopedia and Dictionary of Medicine, Nursing, and Allied Health, Seventh Edition. © 2003 by Saunders, an imprint of Elsevier, Inc. All rights reserved.

den·dri·form

(den'dri-fōrm),
Tree-shaped, or branching.
Synonym(s): arborescent, dendritic (1) , dendroid
[G. dendron, tree, + L. forma, form]
Farlex Partner Medical Dictionary © Farlex 2012

dendriform

(dĕn′drə-fôrm′)
adj.
Shaped like or having the form of a tree.
The American Heritage® Medical Dictionary Copyright © 2007, 2004 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved.

den·dri·form

(den'dri-fōrm)
Tree-shaped, or branching.
Synonym(s): arborescent, dendritic (1) , dendroid.
[G. dendron, tree, + L. forma, form]
Medical Dictionary for the Health Professions and Nursing © Farlex 2012
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Let us first recall that a dendriform algebra [8, 9, 10, 5] is a family (A, [??], [??]) such that A is a vector space and [??], [??] are two products on A, satisfying three axioms.
Definition 4.1 (Definition 2 of [5]) A dendriform coalgebra is a family (C, [[DELTA].sub.[??]], [[DELTA].sub.[??]]) such that:
Proposition 4.2 The triplet ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a dendriform coalgebra.
Indeed, the necessary compatibilities for the dendriform coalgebra ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) to be a codendriform bialgebra [5] write:
Dendriform structures for restriction-deletion and restriction-contraction matroid Hopf algebras
It is itself a subalgebra of WQSym, based on packed words (or set compositions), in which the role of PBT is played by the free dendriform trialgebra on one generator TD (based on Schroder trees), the free cubical trialgebra TC (segmented compositions).
The realization of the free dendriform trialgebra given in [11] involves the following construction.
CHAPOTON, Some dendriform functors, arXiv:0909.2751.
The # product in combinatorial Hopf algebras
Baxter is free as dendriform algebra on its totally primitive elements.
Algebraic and combinatorial structures on Baxter permutations

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