Algebraic properties of overflow semirings

Abstract

We introduce the overflow semiring $S=A{\uplus }_{\mathrm{ord}}L$, extending a positive information algebra $A$ by a join-semilattice $L$, where elements of $L$ dominate $A$ and arithmetic in $L$ reduces to the join. This models overflow in computational systems and generalizes the transition from finite to infinite cardinal arithmetic. We characterize the idempotent elements of $S$ and $S[X]$, fully classify idempotent power series over cardinal numbers, describe the structure of prime and maximal ideals, compute the Krull dimension of $S$ ($\dim S=\dim A+|L|$ for well-ordered finite $L$), and establish Noetherian and Artinian criteria.

arXiv commentMajor revision. Comments still welcomed