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 .
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