AI 中文总结
本文研究由整除序定义的半环的有限基性质，给出了幂族与线性族的有限基判定准则，并解决了相关的极小性问题。
AI 中文摘要
我们研究由交换词通过对其子词赋予整除序而得到的加法幂等半环。与单个字母的幂相关联的每个有限半环都是有限基的，而与线性词相关联的有限半环当且仅当该词长度至多为2时是有限基的。一个超图保持引理给出了非有限基的结果，并将其推广到簇的区间。我们在幂族与线性族之间建立了严格的包含准则，并确定了由每个族的一个成员生成的任意两个簇的并的有限基性质。无限制的幂族生成非有限基的max-plus簇。相比之下，无限制的线性族具有有限基，且其与每个有限幂成员的并也具有有限基。我们还实现了先前已知的六元极限半环作为八元线性词半环的子半环的商。这给出了一个真非有限基子簇，并解决了相应的极小性问题。
英文摘要
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.
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