The finite basis problem and subvariety intervals for max-plus matrix semirings

Abstract

Let $S=(\mathbb{N},\max ,+)$ be the max-plus algebra of the nonnegative integers, and let $\overline{S}$ be obtained from $S$ by adjoining $-\infty$ as a semiring zero. We first prove that the varieties generated by the full matrix semirings $\Mat_n(S)$ and by the upper triangular matrix semirings $\UT_n(\overline S)$ form strictly ascending chains. Every interval between distinct members of either chain has cardinality $2^{{\aleph }_0}$ and contains a chain and an antichain of that cardinality. The proof uses finite syntactic semirings associated with prime-cycle languages and an independent family of identities. We then construct finite commutative ai-semirings from languages of monomials closed under divisibility and prove their bounded-variable property by cancellation, explicit integer evaluations and independence in a free abelian group. A loop-block identity for upper triangular matrices and dimension-independent cubic-padding identities for full matrices show that every variety in each of the above intervals has infinite axiomatic rank. In fact, every variety between $\V(S)$ and the variety generated by all full matrix dimensions has infinite axiomatic rank. These varieties are therefore nonfinitely based, also relative to the ai-semiring axioms. Finally, for every fixed full matrix dimension, we prove equational equivalence over all six positive or nonnegative integer, rational and real coefficient domains. The original convex-ideal countermodels and their full-matrix transfer proof are recalled separately for comparison; they are not used in the algebraic proofs. Throughout, the signature consists of two binary operations and contains no constant symbols.