# Gr\"obner--Shirshov bases for commutative dialgebras

## TL;DR

This paper develops a Gr"obner--Shirshov bases theory for commutative dialgebras, providing normal forms, solvability of the word problem, and basis construction methods, extending algebraic computational tools to this class of algebras.

## Contribution

It introduces a unique reduced Gr"obner--Shirshov basis theory for commutative dialgebras, including finite basis existence and applications to word problems and ideal equality.

## Key findings

- Unique reduced Gr"obner--Shirshov bases exist for ideals in commutative dialgebras.
- The word problem for finitely presented commutative dialgebras is solvable.
- Deciding ideal equality is solvable when the generating set is finite.

## Abstract

We establish Gr\"obner--Shirshov bases theory for commutative dialgebras. We show that for any ideal $I$ of $Di[X]$, $I$ has a unique reduced Gr\"obner--Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set $X$, in particular, $I$ has a finite Gr\"obner--Shirshov basis if $X$ is finite. As applications, we give normal forms of elements of an arbitrary commutative disemigroup, prove that the word problem for finitely presented commutative dialgebras (disemigroups) is solvable, and show that if $X$ is finite, then the problem whether two ideals of $Di[X]$ are identical is solvable. We construct a Gr\"obner--Shirshov basis in associative dialgebra $Di\langle X\rangle$ by lifting a Gr\"obner--Shirshov basis in $Di[X]$.