Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Free Rota-Baxter family algebras and Free (tri)dendriform family
  algebras   In this paper, we first construct the free Rota-Baxter family algebra
generated by some set $X$ in terms of typed angularly $X$-decorated planar
rooted trees. As an application, we obtain a new construction of the free
Rota-Baxter algebra only in terms of angularly decorated planar rooted trees
(not forests), which is quite different from the known construction via
angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We
then embed the free dendriform (resp. tridendriform) family algebra into the
free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove
that the free Rota-Baxter family algebra is the universal enveloping algebra of
the free (tri)dendriform family algebra.

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The categories of this paper are: nlin
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The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th