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

TL;DR

This paper develops Gr"obner-Shirshov bases for $L$-algebras, providing normal forms, embedding theorems, and applications to free dialgebras and free products, advancing the algebraic theory of $L$-algebras.

Contribution

It establishes Composition-Diamond lemmas for $ ext{Omega}$-algebras and $L$-algebras, introduces Gr"obner-Shirshov bases, and proves embedding theorems for $L$-algebras.

Findings

Normal form for free $L$-algebra obtained

Embedding theorems for $L$-algebras proved

Gr"obner-Shirshov bases constructed for free dialgebra and free product

Abstract

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