Formal Concepts and Residuation on Multilattices

Multilattices are generalisations of lattices introduced by Mihail Benado. Hereplaced the existence of unique lower (resp. upper) bound by the existence ofmaximal lower (resp. minimal upper) bound(s). A multilattice will be calledpure 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 fuzzysetting. In this paper we exhibit the smallest pure multilattice and show thatit is a sub-multilattice of any pure multilattice. We also prove that anybounded residuated multilattice that is not a residuated lattice has at leastseven elements. We apply the ordinal sum construction to get more examples ofresiduated multilattices that are not residuated lattices. We then use theseresiduated multilattices to evaluate objects and attributes in formal conceptanalysis setting, and describe the structure of the set of corresponding formalconcepts. More precisely, if $\mathcal{A}_i:=(A_i,\le_i,\top_i,\odot_i,\to_i,\bot_i)$, $i=1,2$ are two complete residuatedmultilattices, $G$ and $M$ two nonempty sets and $(\varphi, \psi)$ a Galoisconnection between $A_1^G$ and $A_2^M$ that is compatible with the residuation,then we show that \[\mathcal{C}: =\{(h,f)\in A_1^G\times A_2^M; \varphi(h)=f \text{ and }\psi(f)=h \}\] can be endowed with a complete residuated multilatticestructure. This is a generalization of a result by Ruiz-Calvi{\~n}o and Medinasaying that if the (reduct of the) algebras $\mathcal{A}_i$, $i=1,2$ arecomplete multilattices, then $\mathcal{C}$ is a complete multilattice.Comment: 21 pages, 6 figure