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.