Gröbner–Shirshov bases for commutative dialgebras

Abstract

We establish Gröbner–Shirshov bases theory for commutative dialgebras. We show that for any ideal I of $Di[X]$, I has a unique reduced Gröbner–Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set X, in particular, I has a finite Gröbner–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 Gröbner–Shirshov basis in associative dialgebra $Di〈X〉$ by lifting a Gröbner–Shirshov basis in $Di[X]$.

Keywords:

• Commutative dialgebra
• commutative disemigroup
• Gröbner–Shirshov basis
• normal form
• word problem

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A99
• 16S15
• 13P10
• 08A50

Acknowledgements

We wish to express our thanks to the referee for helpful suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by the National Natural Science Foundation of China (11571121), the Natural Science Foundation of Guangdong Province (2017A030313002) and the Science and Technology Program of Guangzhou (201707010137).