c0semirings

c0semirings are a class of commutative semirings that arise naturally in the study of continuous functions on locally compact Hausdorff spaces and in the theory of Banach spaces of sequences. The designation “c0’’ refers to the classical notation for the space of real or complex sequences that converge to zero; it also appears in the notation C0(X) for the algebra of continuous complex‑valued functions on a locally compact space X that vanish at infinity. A c0semiring is defined as a commutative semiring (S,+,·,0,1) equipped with a topology that makes S into a topological semiring and such that for every element s∈S there exists a sequence (sn) in S with sn→0 and s=sn a, where a is a distinguished idempotent element. This topological condition ensures that the semiring behaves like a Banach space of functions that vanish at infinity.

Typical examples are the semirings of continuous functions C0(X) endowed with pointwise addition and multiplication, where X is a locally compact Hausdorff space. In the sequence setting, the c0 space of convergent‑to‑zero sequences with component‑wise operations is a c0semiring. The algebraic operations are associative, commutative, and distributive, and the presence of the topological “c0’’ property guarantees the Archimedean completeness of the underlying semiring.

C0semirings play a role in functional analysis, particularly in the study of C*‑algebras, because the C0(X) semiring can be embedded into the complex C*‑algebra of continuous functions vanishing at infinity. They also provide a convenient framework for topics such as K‑theory of C*‑algebras, the representation theory of topological groups, and the modeling of signal processing systems where “vanishing at infinity’’ corresponds to decay conditions.

For further details on the theory of c0semirings, see Banach space textbooks that cover the c0 sequence space, as well as research articles on semirings in functional analysis.