Finite bases and joins for semirings defined by the divisibility order

Abstract

We study additively idempotent semirings obtained from commutative words by equipping their subwords with the divisibility order. Every finite semiring associated with a power of one letter is finitely based, whereas one associated with a linear word is finitely based exactly when the word has length at most two. A hypergraph preservation lemma yields the nonfinite basis result and extends it to intervals of varieties. We establish a sharp containment criterion between the power and linear families, and determine the finite basis property of every join of two varieties generated by one member of each family. The unrestricted power family generates the nonfinitely based max-plus variety. In contrast, the unrestricted linear family has a finite basis, as does its join with each finite power member. We also realize a previously known six-element limit semiring as a quotient of a subsemiring of the eight-element linear-word semiring. This gives a proper nonfinitely based subvariety and resolves the corresponding minimality question.