Formal Concepts and Residuation on Multilattices

Multilattices are generalisations of lattices introduced by Mihail Benado in [4]. 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, i

Formal Concepts and Residuation on Multilattices}

Let $\mathcal{A}_i: =(A_i,\le_i,\top_i,\odot_i,\to_i,\bot_i)$, $i=1,2$ be two complete residuated multilattices, $G$ (set of objects) and $M$ (set of attributes) be two nonempty sets and $(\varphi, \psi)$ a Galois connection between $A_1^G$ and $A_2^M$. In this work we prove that $$\mathcal{C}: =\{(h,f)\in A_1^G\times A_2^M \mid \varphi(h)=f \text{ and } \psi(f)=h \}$$ is a complete residuated multilattice. This is a generalization of a result by Ruiz-Calvi{\~n}o and Medina \cite{RM12} saying that if the (reduct of the) algebras $\mathcal{A}_i$, $i=1,2$ are complete multilattices, then $\mathcal{C}$ is a complete multilattice.Comment: 14 pages, 3 figure

Weak-hyperlattices derived from fuzzy congruences

In this paper we explore the connections between fuzzy congruence relations, fuzzy ideals and homomorphisms of hyperlattices. Indeed, we introduce the concept of fuzzy quotient set of hyperlattices as it was done in the case of rings [19]. We prove that a fuzzy congruence induces a fuzzy ideal of the fuzzy quotient hyperlattice. In particular, we establish necessary and sufficient conditions for a zero-fuzzy congruence class to be a fuzzy ideal of a hyperlattice

ℒ-Fuzzy Ideals of Residuated Lattices

This paper mainly focuses on building the ℒ-fuzzy ideals theory of residuated lattices. Firstly, we introduce the notion of ℒ-fuzzy ideals of a residuated lattice and obtain their properties and equivalent characterizations. Also, we introduce the notion of prime fuzzy ideal, fuzzy prime ideal and fuzzy prime ideal of the second kind of a residuated lattice and establish existing relationships between these types of fuzzy ideals. Finally, we investigate the notions of fuzzy maximal ideal and maximal fuzzy ideal of a residuated lattice and present some characterizations