论文标题
多弹药的正式概念和差异
Formal Concepts and Residuation on Multilattices
论文作者
论文摘要
多层是Mihail Benado引入的晶格的概括。他取代了由最大下部（分别最小上部）结合的独特的下部（分别上部）结合的独特的存在。如果不是晶格，则多弹性将被称为纯净。多层可以赋予残留物，因此用作评估模糊环境中元素的一组真实价值。在本文中，我们展示了最小的纯多弹性，并表明它是任何纯多弹药的亚ultilattice。我们还证明，任何不是残留晶格的有界残留的多弹药都有至少七个元素。我们应用序数构造，以获取更多不属于残留晶格的残留多弹药的示例。然后，我们使用这些残留的多弹药来评估正式概念分析设置中的对象和属性，并描述相应形式概念集的结构。更确切地说，如果$ \ nathcal {a} _i：=（a_i，\ le_i，\ top_i，\ odot_i，\ odot_i，\ to_i，\ to_i，\ bot_i）$，$ i = 1,2 $是两个完整的残留的多层次多ilattices，$ g $和$ m $ m $两个非exprent $ $ m $ $ $ a_ $（ $ a_2^m $与残留物兼容，然后我们证明 \ [\ Mathcal {C}：= \ {（h，f）\ in A_1^g \ times a_2^m; φ（h）= f \ text {and}ψ（f）= h \} \]可以用完整的残基多弹性结构来赋予。这是Ruiz-Calvi {ñ} O和Medina对结果的概括，如果（减少）代数$ \ Mathcal {a} _i $，$ i = 1,2 $是完整的多弹药，那么$ \ nathcal {c} $是完整的多层次。
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.
