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arXiv · 1907.06680

Gr\"obner--Shirshov bases for commutative dialgebras

Yuqun Chen·Guangliang Zhang
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Abstract

We establish Gr\"obner--Shirshov bases theory for commutative dialgebras. We show that for any ideal $I$ of $Di[X]$, $I$ has a unique reduced Gr\"obner--Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set $X$, in particular, $I$ has a finite Gr\"obner--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 Gr\"obner--Shirshov basis in associative dialgebra $Di\langle X\rangle$ by lifting a Gr\"obner--Shirshov basis in $Di[X]$.

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Yuqun Chen, Guangliang Zhang. 2019-07-11. Gr\"obner--Shirshov bases for commutative dialgebras. https://doi.org/10.1080/00927872.2018.1513017

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