AI 中文总结
本文证明非负整数上max-plus簇有唯一极限子簇，刻画其遗传有限基性，并证明真子簇局部有限及有限截断的有限基性。
AI 中文摘要
我们证明了由非负整数上的max-plus代数生成的簇具有唯一的极限子簇，即由Shao、Ren和Gao构造的六元素半环所生成的簇。更一般地，我们刻画了包含max-plus簇的交换加性幂等半环的有限定义簇中的遗传有限基性。其遗传有限基子簇构成一个有限格，并允许涉及至多十个变量的等式基。我们还证明了max-plus簇的每个真子簇都是局部有限的。关于具有乘法单位元的有限半环的有限基定理确立了所有有限截断的有限基性。最后，我们确定了由幂等式定义的两个子簇族以及由任意等式族相对定义的、使单个加性子项成为最大元素的子簇的有限基性质。
英文摘要
We prove that the variety generated by the max-plus algebra on the nonnegative integers has a unique limit subvariety, namely the variety generated by the six-element semiring constructed by Shao, Ren and Gao. More generally, we characterize hereditary finite basedness in a finitely defined variety of commutative additively idempotent semirings containing the max-plus variety. Its hereditarily finitely based subvarieties form a finite lattice and admit equational bases involving at most ten variables. We also prove that every proper subvariety of the max-plus variety is locally finite. A finite basis theorem for finite semirings with a multiplicative identity establishes finite basedness of all finite truncations. Finally, we determine the finite basis properties of two families of subvarieties defined by power identities and of subvarieties relatively defined by arbitrary families of identities making individual additive subterms greatest elements.
