Further Generalizations
A near-ring does not require addition to be commutative, nor does it require right-distributivity. Just as cardinal numbers form a semiring, so do ordinal numbers form a near-ring.
In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig.
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