Equational theories of min-plus semirings

Abstract

We study the equational theory and subvarieties of the variety $\mathcal V$ generated by $(\mathbb N_0,\min,+)$. We prove that a subvariety of $\mathcal V$ is proper if and only if it is locally finite, equivalently, if and only if it satisfies a power-stabilization identity. Every finite member generates a hereditarily finitely based variety with a finite subvariety lattice. In contrast, the periodic subvarieties defined by $x^m=x^{m+1}$ are nonfinitely based for $m\geq2$. We construct a strictly descending chain of nonfinitely based subvarieties and three varieties generated by explicit reduced products, each with a decidable equational theory. The smallest is a limit variety: it is nonfinitely based, and every proper subvariety has an absolute finite identity basis. The subvariety lattice of $\mathcal V$ has cardinality continuum. In fact, the interval between the first reduced-product variety and the subvariety defined by $x^2=x^3$ already has this cardinality, and all its members are nonfinitely based. We also give a hereditarily finitely based example with an infinite subvariety lattice. The constructions use finite convex models and recognition quotients.