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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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