# A new Composition-Diamond lemma for dialgebras

## TL;DR

This paper introduces a new Composition-Diamond lemma for dialgebras using a more flexible ordering, establishing an equivalence between Gr"obner-Shirshov bases and linear bases, and providing practical normal form methods.

## Contribution

It presents a more general and convenient Composition-Diamond lemma for dialgebras, ensuring the uniqueness of reduced Gr"obner-Shirshov bases and improving normal form computations.

## Key findings

- New Composition-Diamond lemma with arbitrary monomial-center ordering
- Equivalence of Gr"obner-Shirshov basis and linear basis conditions
- Method for normal forms of elements in disemigroups

## Abstract

Let $Di\langle X\rangle$ be the free dialgebra over a field generated by a set $X$. Let $S$ be a monic subset of $Di\langle X\rangle$. A Composition-Diamond lemma for dialgebras is firstly established by Bokut, Chen and Liu in 2010 \cite{Di} which claims that if (i) $S$ is a Gr\"{o}bner-Shirshov basis in $Di\langle X\rangle$, then (ii) the set of $S$-irreducible words is a linear basis of the quotient dialgebra $Di\langle X \mid S \rangle$, but not conversely. Such a lemma based on a fixed ordering on normal diwords of $Di\langle X\rangle$ and special definition of composition trivial modulo $S$. In this paper, by introducing an arbitrary monomial-center ordering and the usual definition of composition trivial modulo $S$, we give a new Composition-Diamond lemma for dialgebras which makes the conditions (i) and (ii) equivalent. We show that every ideal of $Di\langle X\rangle$ has a unique reduced Gr\"{o}bner-Shirshov basis. The new lemma is more useful and convenient than the one in \cite{Di}. As applications, we give a method to find normal forms of elements of an arbitrary disemigroup, in particular, A.V. Zhuchok's (2010) and Y.V. Zhuchok's (2015) normal forms of the free commutative disemigroups and the free abelian disemigroups, and normal forms of the free left (right) commutative disemigroups.