Abstract
We establish Gröbner–Shirshov bases theory for commutative dialgebras. We show that for any ideal I of , I has a unique reduced Gröbner–Shirshov basis, where
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
are identical is solvable. We construct a Gröbner–Shirshov basis in associative dialgebra
by lifting a Gröbner–Shirshov basis in
.
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.