# Sets with two associative operations

Open Mathematics (2003)

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

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topTeimuraz 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

top- [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] J.-L. Loday: “Algébras ayant deux opérations associatives (digébres)”, C. R. Acad. Sci. Paris Sér. I Math., Vol. 321, (1995), pp. 141–146.
- [3] J.-L. Loday: “Dialgebras”, In: Dialgebras and related operads, Lecture Notes in Math. 1763, Springer, Berlin, 2001, pp. 7–66. Zbl0999.17002
- [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] 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] J.-L. Loday and M. O. Ronco: Trialgebras and families of polytopes, Preprint (2002), http://front.math.ucdavis.edu/math.AT/0205043.
- [7] B. Richter: Dialgebren, Doppelalgebren und ihre Homologie, Diplomarbeit, Universität Bonn, 1997, http://www.math.uni-bonn.de/people/richter/.

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