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Gröbner-Shirshov bases for commutative dialgebras. (English) Zbl 07098068
Summary: 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 $$Di\langle X\rangle$$ by lifting a Gröbner-Shirshov basis in $$Di[X]$$.
##### MSC:
 17A99 General nonassociative rings 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 08A50 Word problems (aspects of algebraic structures)
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