GRÖBNER–SHIRSHOV BASES FOR L-ALGEBRAS

Abstract

In 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.