Abstract
Bracketed words are basic structures both in mathematics (such as Rota-Baxter algebras) and mathematical physics (such as rooted trees) where the locations of the substructures are important. In this paper, we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested, and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths.
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References
Alonso L. Uniform generation of a Motzkin word. Theoret Comput Sci, 1994, 134: 529–536
Baader F, Nipkow T. Term Rewriting and All That. Cambridge: Cambridge Univ Press, 1998
Bogner C, Weinzierl S. Blowing up Feynman integrals. Nucl Phys B Proc Suppl, 2008, 183: 256–261
Bokut L A, Chen Y Q, Chen Y S. Composition-Diamond lemma for tensor product of free algebras. J Algebra, 2010, 323: 2520–2537
Bokut L A, Chen Y Q, Deng X, Gröbner-Shirshov bases for Rota-Baxter algebras. Sib Math J, 2010, 51: 978–988
Bokut L A, Chen Y Q, Li Y. Gröbner-Shirshov bases for categories. In: Operads and Universal Algebra. Singapore: World Scientific Press, 2012, 1–23
Bokut L A, Chen Y Q, Qiu J. Greobner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras. J Pure Appl Algebra, 2010, 214: 89–110
Buchberger B. An algorithm for finding a basis for the residue class ring of a zerodimensional polynomial ideal. Ph D Thesis, University of Innsbruck, Austria, 1965 (in German)
Connes A, Kreimer D. Hopf algebras, renormalization and concommutative geometry. Comm Math Phys, 1998, 199: 203–242
Connes A, Kreimer D. Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm Math Phys, 2000, 210: 249–273
Donaghey R, Shapiro L W. Motzkin numbers. J Combin Theory Ser A, 1977, 23: 291–301
Flajolet P. Mathematical methods in the analysis of algorithms and data structures. In: Trends in Theoretical Computer Science (Udine, 1984). Principles Comput Sci Ser 12. Rockville: Computer Sci Press, 1988, 225–304
Guo L. Operated semigroups, Motzkin paths and rooted trees. J Algebraic Combin, 2009, 29: 35–62
Guo L. An Introduction to Rota-Baxter Algebra. Beijing/Boston: Higher Education Press/International Press, 2012
Guo L, Sit W, Zhang R. Differential type operators and Gröbner-Shirshov bases. J Symbolic Comput, 2013, 52: 97–123
Krajewski T, Wulkenhaar R. On Kreimer’s Hopf algebra structure of Feynman graphs. Eur Phys J C, 1999, 7: 697–708
Kreimer D. On overlapping divergences. Comm Math Phys, 1999, 204: 669–689
Sapounakis A, Tsikouras P. On k-colored Motzkin words. J Integer Seq, 2004, 7: Article 04. 2.5.
Shirshov A I. Some algorithmic problem for Å-algebras. Sibirsk Mat Zh, 1962, 3: 132–137
Zheng S, Gao X, Guo L, Sit W. Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases. Preprint
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Zheng, S., Guo, L. Relative locations of subwords in free operated semigroups and Motzkin words. Front. Math. China 10, 1243–1261 (2015). https://doi.org/10.1007/s11464-014-0379-1
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DOI: https://doi.org/10.1007/s11464-014-0379-1