Sets with two associative operations

Teimuraz Pirashvili

Open Mathematics (2003)

  • Volume: 1, Issue: 2, page 169-183
  • ISSN: 2391-5455

Abstract

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In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.

How to cite

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Teimuraz Pirashvili. "Sets with two associative operations." Open Mathematics 1.2 (2003): 169-183. <http://eudml.org/doc/268931>.

@article{TeimurazPirashvili2003,
abstract = {In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.},
author = {Teimuraz Pirashvili},
journal = {Open Mathematics},
keywords = {05E99; 20M05; 08B20},
language = {eng},
number = {2},
pages = {169-183},
title = {Sets with two associative operations},
url = {http://eudml.org/doc/268931},
volume = {1},
year = {2003},
}

TY - JOUR
AU - Teimuraz Pirashvili
TI - Sets with two associative operations
JO - Open Mathematics
PY - 2003
VL - 1
IS - 2
SP - 169
EP - 183
AB - In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.
LA - eng
KW - 05E99; 20M05; 08B20
UR - http://eudml.org/doc/268931
ER -

References

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  1. [1] M. Aguiar and F. Sottile: Structures of the Malvenuto-Reutenauer Hopf algebra of permutations, Preprint (2002), http://front.math.ucdavis.edu/math.CO/0203282. 
  2. [2] J.-L. Loday: “Algébras ayant deux opérations associatives (digébres)”, C. R. Acad. Sci. Paris Sér. I Math., Vol. 321, (1995), pp. 141–146. 
  3. [3] J.-L. Loday: “Dialgebras”, In: Dialgebras and related operads, Lecture Notes in Math. 1763, Springer, Berlin, 2001, pp. 7–66. Zbl0999.17002
  4. [4] J.-L. Loday: “Arithmetree”, J. of Algebra, Vol. 258, (2002), pp. 275–309. http://dx.doi.org/10.1016/S0021-8693(02)00510-0 Zbl1063.16044
  5. [5] J.-L. Loday and M. O. Ronco: “Order structure on the algebra of permutations and of planar binary trees”, J. Algebraic Combin., Vol. 15, (2002), pp. 253–270 http://dx.doi.org/10.1023/A:1015064508594 Zbl0998.05013
  6. [6] J.-L. Loday and M. O. Ronco: Trialgebras and families of polytopes, Preprint (2002), http://front.math.ucdavis.edu/math.AT/0205043. 
  7. [7] B. Richter: Dialgebren, Doppelalgebren und ihre Homologie, Diplomarbeit, Universität Bonn, 1997, http://www.math.uni-bonn.de/people/richter/. 

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