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TitleGr\"obner-Shirshov bases for $L$-algebras.
Author(s) Leonid A. Bokut, Yu-Fu Chen, Jiapeng Huang
TypeArticle in Journal
AbstractIn this paper, we first establish Composition-Diamond lemma for Ω-algebras. We give a Gröbner–Shirshov basis of the free L-algebra as a quotient algebra of a free Ω-algebra, and then the normal form of the free L-algebra is obtained. Second we establish Composition-Diamond lemma for L-algebras. As applications, we give Gröbner–Shirshov bases of the free dialgebra and the free product of two L-algebras, and then we show four embedding theorems of L-algebras: (1) Every countably generated L-algebra can be embedded into a two-generated L-algebra. (2) Every L-algebra can be embedded into a simple L-algebra. (3) Every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. (4) Three arbitrary L-algebras A, B, C over a field k can be embedded into a simple L-algebra generated by B and C if |k| ≤ dim(B * C) and |A| ≤ |B * C|, where B * C is the free product of B and C.

KeywordsGröbner–Shirshov basis; Ω-algebra; dialgebra; L-algebra
ISSN0218-1967; 1793-6500/e
URL http://www.worldscientific.com/doi/abs/10.1142/S0218196713500094
JournalInt. J. Algebra Comput.
PublisherWorld Scientific, Singapore
Translation No
Refereed No