Positive rigs

Abstract A positive rig is a commutative and unitary semi-ring A such that 1 + x {1+x} is invertible for every x ∈ A {x\in A} . We show that the category of positive rigs shares many properties with that of K -algebras for a (non-algebraically closed) field K . In particular, it is coextensive and, although we do not have an analogue of Hilbert’s basis theorem for positive rigs, we show that every finitely presentable positive rig is a finite direct product of directly indecomposable ones. We also describe free positive rigs as rigs of rational functions with non-negative rational coefficients, and we give a characterization of the positive rigs with a unique maximal ideal.