Gr\"obner-Shirshov bases for $L$-algebras

In this paper, we firstly establish Composition-Diamond lemma for $\mathrm{\Omega }$-algebras. We give a Gr"{o}bner-Shirshov basis of the free $L$-algebra as a quotient algebra of a free $\mathrm{\Omega }$-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 Gr"{o}bner-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 $\mathrm{\mid }k\mathrm{\mid }\le \dim (B\ast C)$ and $\mathrm{\mid }A\mathrm{\mid }\le \mathrm{\mid }B\ast C\mathrm{\mid }$, where $B\ast C$ is the free product of $B$ and $C$.