Gr\"obner-Shirshov bases for semirings

Abstract

In the paper, we establish Gr\"obner-Shirshov bases for semirings and commutative semirings. As applications, we obtain Gr\"obner-Shirshov bases and A. Blass's (1995) and M. Fiore -T. Leinster's (2004) normal forms of the semirings $\mathbb{N}[x]/(x=1+x+x^2)$ and $\mathbb{N}[x]/(x=1+x^2)$ with one generator $x$ and one defining relation, correspondingly.