Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is $\overline{\mathcal{K}}$-four-fold ($\overline{\mathcal{K}}$-4FCA ) where $\overline{\mathcal{K}}$ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of $\overline{\mathcal{K}}$-four-fold that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-à-vis previous formal conceptual analyses and some affordances of their results.

中文翻译：

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基于完全幂等半场的四重形式概念分析

形式概念分析（FCA）是一种以格子和阶理论为基础的著名的监督型布尔数据挖掘技术，例如对模糊环和幂等半环具有若干扩展。FCA的核心是两个动力装置之间的Galois连接。在本文中，我们扩展了FCA形式主义，以同时包括幂等半场上四个不同半向量空间之间的所有四个Galois连接。结果是$\overline{\mathcal{ķ}}$-四折（$\overline{\mathcal{ķ}}$-4FCA）其中 $\overline{\mathcal{ķ}}$是幂等半场偏向分析。由于完整的幂等半场成对出现（例如，完整的最大正半圆环和最小正半圆环），因此基本构造显示了由双阶，行，列和伽罗瓦连接引起的对偶性，这些对偶性同时出现在多个时间以提供完整的可变性。我们的结果导致了一个基本定理$\overline{\mathcal{ķ}}$-四倍，正确地将四元组定义为矢量的（顺序）同构晶格的四元组，并讨论了其与先前正式概念分析的相关性以及其结果的某些可承受性。