Supernilpotent Semirings
Nebojša Mudrinski, Milica Šobot
TL;DR
The paper investigates the relationship between supernilpotency and nilpotency in semirings with absorbing zero by extending higher-arity commutator theory to this setting. It establishes a necessary and sufficient condition for both properties: a semiring $\mathbf{S}=(S,+,\cdot,o)$ is $n$-nilpotent iff it is additively cancellative and $S^{n+1}=\{o\}$, which is also equivalent to being $n$-supernilpotent. A congruence- and ideal-theoretic framework is developed, showing that in additively cancellative semirings the binary commutator matches the induced ideal-congruence, with corollaries characterizing abelian and solvable structures and, in the finite case, reducing to finite rings. These results unify nilpotency notions in semirings and provide a practical criterion for detecting supernilpotency, with implications for the study of algebraic properties defined by commutators in semiring-like structures.
Abstract
We prove that supernilpotent and nilpotent semirings with absorbing zero are the same and provide a necessary and sufficient condition for supernilpotency (nilpotency).