发表机构
Chongqing University of Technology（重庆理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究四阶半环的有限基问题，结合现有分类与结构约简，在2341个类型中确定2284个有限基和57个非有限基类型，并给出完整目录与对应表。
AI 中文摘要
小阶半环的有限基问题与其半群对应问题不同，即使在阶数为三时也是如此。近期工作对四元素加法幂等半环的若干加法类型进行了分类，包括所有加法约简为链的386个半环。我们考虑所有具有交换加法的四元素半环，采用不含命名常数的二元签名。结合现有分类、结构约简与多项式范式，我们在2341个同构类型中获得了2284个有限基类型和57个非有限基类型。非有限基情形包括45个加法幂等半环和12个加法非幂等半环。肯定性论证给出了显式基或具有指定界限的有限构造。否定性论证使用了引用结果、项收缩和超图障碍。一个完整目录记录了每个代表所适用的结果；单独的对应表精确标识了由先前分类提供的案例。
英文摘要
The finite basis problem for small semirings differs from its semigroup counterpart even in order three. Recent work classifies several additive types of four-element additively idempotent semirings, including all 386 semirings whose additive reduct is a chain. We consider all four-element semirings with commutative addition, in the binary signature without named constants. Combining the existing classifications with structural reductions and polynomial normal forms, we obtain 2284 finitely based and 57 nonfinitely based isomorphism types among the 2341 types. The nonfinitely based cases consist of 45 additively idempotent semirings and twelve with nonidempotent addition. The positive arguments give explicit bases or finite constructions with specified bounds. The negative arguments use cited results, term retractions and a hypergraph obstruction. A complete catalogue records the applicable result for each representative; separate correspondence tables identify precisely the cases supplied by the earlier classifications.
