AI 中文总结
研究半环中减法闭包算子在理想运算上的分配性，引入库拉托夫斯基半环和核半环，给出反例与正面结果，如指出$\mathbb{N}_0$等是核半环，$T = [0,1]$是库拉托夫斯基半环等，还纠正相关断言。
AI 中文摘要
本文研究交换半环理论中的减法闭包算子。该算子虽满足标准闭包性质，但一般不在理想加法、交或乘法上分配。我们引入了库拉托夫斯基半环（闭包保持并）和核半环（闭包保持交），给出了各自的反例和正面结果。主要发现包括：标准半环$\mathbb{N}_0$和最大加代数是具有乘法分配性的核半环；区间半环$T = [0,1]$是库拉托夫斯基且乘法分配但非核半环；零和自由半环上的多项式半环$S_0[X,Y,Z]$构成一大类非核反例。我们还纠正了近期关于任意交分配性的一个断言。
英文摘要
In this paper, we investigate the subtractive closure operator in commutative hemiring theory. While this operator satisfies standard closure properties, it does not distribute over ideal addition, intersection, or multiplication in general. We introduce Kuratowski hemirings (where closure preserves joins) and nucleus hemirings (where closure preserves meets), providing explicit counterexamples and positive results for each. Key findings include: the standard semiring $\mathbb{N}_0$ and the max-plus algebra are nucleus hemirings with multiplicative distributivity; the interval semiring $T = [0,1]$ is Kuratowski and multiplicatively distributive but not nucleus; and polynomial hemirings $S_0[X,Y,Z]$ over zerosumfree hemirings form a broad class of non-nucleus counterexamples. We also correct a recent claim regarding distributivity over arbitrary intersections.
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