dendriform algebra
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Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 190
Author(s):  
Yi Zhang ◽  
Xiaosong Peng ◽  
Yuanyuan Zhang

As a generalization of Rota–Baxter algebras, the concept of an Ω-Rota–Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an Ω-dendriform algebra and show the relationship between Ω-Rota–Baxter algebras and Ω-dendriform algebras. Then, we provide a multiplication recursion definition of typed, angularly decorated rooted trees. Finally, we construct the free Ω-Rota–Baxter algebra by typed, angularly decorated rooted trees.


2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Vincent Vong

International audience We present some results on the freeness or non freeness of some tridendriform algebras. In particular, we give a combinatorial proof of the freeness of WQSym, an algebra based on packed words, result already known with an algebraic proof. Then, we prove the non-freeness of an another tridendriform algebra, PQSym, a conjecture remained open. The method of these proofs is generalizable, in particular it has been used to prove the freeness of the dendriform algebra FQSym and the quadrialgebra of 2-permutations.


2019 ◽  
Vol 12 (3) ◽  
pp. 734-748
Author(s):  
Sabri Kaan Gürbüzer ◽  
Bedia Akyar

We determine the triangulation of a cyclohedron compatible with the Tamari orderon its faces. We dene a name of a tubing on a path and a plumbing leading us to construct the dendriform algebra of the collection of maximal tubings on paths. Moreover, we give an operad structure of associahedra and a module structure of cyclohedra via tubings. We dene labelled maximal tubings on paths and give to an application of tubings in homological algebra.


2017 ◽  
Vol 24 (01) ◽  
pp. 53-74 ◽  
Author(s):  
Shuyun Zhou ◽  
Li Guo

A dendriform algebra defined by Loday has two binary operations that give a two-part splitting of the associativity in the sense that their sum is associative. Similar dendriform type algebras with three-part and four-part splitting of the associativity were later obtained. These structures can also be derived from actions of suitable linear operators, such as a Rota-Baxter operator or TD operator, on an associative algebra. Motivated by finding a five-part splitting of the associativity, we consider the Rota-Baxter TD (RBTD) operator, an operator combining the Rota-Baxter operator and TD operator, and coming from a recent study of Rota’s problem concerning linear operators on associative algebras. Free RBTD algebras on rooted forests are constructed. We then introduce the concept of a quinquedendriform algebra and show that its defining relations are characterized by the action of an RBTD operator, similar to the cases of dendriform and tridendriform algebras.


2017 ◽  
Vol 69 (1) ◽  
pp. 21-53 ◽  
Author(s):  
Darij Grinberg

AbstractThe dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.


2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Vsevolod Gubarev ◽  
Pavel Kolesnikov

AbstractFollowing a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.


Author(s):  
Philippe Leroux

We propose both a reformulation of some known results on the free dendriform algebra on one generator from a parenthesis setting instead of using permutations and some developments as well. Moreover, by introducing the concept of NCP-operad, we show how to use the free dendriform algebra on one generator to reformulate some results obtained by Speicher in free probability.


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