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GRÖBNER–SHIRSHOV BASES FOR L-ALGEBRAS

L. A. BOKUT, YUQUN CHEN, JIAPENG HUANG

2013
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International journal of algebra and computation
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In this paper, we firstly 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. We secondly 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
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... ebra 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|≤(B*C) and |A|≤|B*C|, where B*C is the free product of B and C.

doi:10.1142/s0218196713500094
fatcat:uxp42pwt5ffyvixlpnjv57qyry