多格中的形式概念与残余运算
Formal Concepts and Residuation on Multilattices
AI总结：
本文研究了多格中的形式概念与残余运算，构造了最小纯多格并证明其为所有纯多格的子多格，同时展示了非残余格的有界残余多格至少有七个元素，并在形式概念分析中应用这些结构。
摘要：
多格是Mihail Benado引入的 lattice 的一般化，他将唯一下界（或上界）的存在替换为最大下界（或最小上界）的存在。多格被称为纯多格如果它不是格。多格可以赋予残余运算，并用于模糊设置中评估元素。本文展示了最小纯多格，并证明其为任何纯多格的子多格。我们还证明，任何非残余格的有界残余多格至少有七个元素。我们应用序和构造来获得更多残余多格的例子。然后使用这些残余多格来评估对象和属性，并描述相应的形式概念集合的结构。更具体地说，如果 $\mathcal{A}_i: =(A_i,\le_i,\top_i,\odot_i,\to_i,\bot_i)$，$i=1,2$ 是两个完整的残余多格，$G$ 和 $M$ 是两个非空集合，$(φ, ψ)$ 是在 $A_1^G$ 和 $A_2^M$ 之间兼容残余运算的伽罗瓦连接，那么我们证明 $\mathcal{C}: =\{(h,f)\in A_1^G\times A_2^M; φ(h)=f \text{ and } ψ(f)=h \}$ 可以赋予完整的残余多格结构。这是对Ruiz-Calvi{ñ}o和Medina结果的推广，后者指出如果代数 $\mathcal{A}_i$，$i=1,2$ 是完整的多格，则 $\mathcal{C}$ 是完整的多格。
英文摘要：
Multilattices are generalisations of lattices introduced by Mihail Benado. He replaced the existence of unique lower (resp. upper) bound by the existence of maximal lower (resp. minimal upper) bound(s). A multilattice will be called pure if it is not a lattice. Multilattices could be endowed with a residuation, and therefore used as set of truth-values to evaluate elements in fuzzy setting. In this paper we exhibit the smallest pure multilattice and show that it is a sub-multilattice of any pure multilattice. We also prove that any bounded residuated multilattice that is not a residuated lattice has at least seven elements. We apply the ordinal sum construction to get more examples of residuated multilattices that are not residuated lattices. We then use these residuated multilattices to evaluate objects and attributes in formal concept analysis setting, and describe the structure of the set of corresponding formal concepts. More precisely, if $\mathcal{A}_i: =(A_i,\le_i,\top_i,\odot_i,\to_i,\bot_i)$, $i=1,2$ are two complete residuated multilattices, $G$ and $M$ two nonempty sets and $(φ, ψ)$ a Galois connection between $A_1^G$ and $A_2^M$ that is compatible with the residuation, then we show that \[\mathcal{C}: =\{(h,f)\in A_1^G\times A_2^M; φ(h)=f \text{ and } ψ(f)=h \}\] can be endowed with a complete residuated multilattice structure. This is a generalization of a result by Ruiz-Calvi{ñ}o and Medina saying that if the (reduct of the) algebras $\mathcal{A}_i$, $i=1,2$ are complete multilattices, then $\mathcal{C}$ is a complete multilattice.
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