In this paper, we discuss and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then go over a specific version of Cramer’s rule which is also related to the pseudo-inverse of the system matrix. In these two methods, the constant vector plays an implicit role in solvability of the system. Another method is called the normalization method in which both the system matrix and the constant vector play explicit roles in the solution process. Each of these methods yields the maximal solution if it exists. Importantly, we show that the maximal solutions obtained from these methods as well as the previously studied LU-method are all identical.

中文翻译：

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幂等半场上的线性系统分析

在本文中，我们讨论并分析了求解幂等半场上线性方程组的方法。第一种方法基于系统矩阵的伪逆。然后，我们遍历Cramer规则的特定版本，该规则也与系统矩阵的伪逆相关。在这两种方法中，常数向量在系统的可解性中起着隐性作用。另一种方法被称为归一化方法，其中系统矩阵和常数向量在求解过程中都扮演着明确的角色。如果存在这些方法，则每种方法都会产生最大解。重要的是，我们表明从这些方法以及先前研究的LU方法获得的最大解都是相同的。