Triadic Concept Analysis (TCA) is an extension of Formal Concept Analysis (FCA) for handling data represented as a set of objects described by attributes and conditions via a ternary relation. However, the intuition to go from FCA to TCA is not always straightforward. In this paper we discuss some FCA notions from dyadic to triadic. Although some ideas admit straightforward adaptation, most do not. In particular, we address the representation problem, the notion of redundant attributes and subcontexts in the triadic setting. • Improve line diagrams (3-nets and nested line diagrams) for triadic data. • Identify reducible attributes in triadic data. • Explore compatibility of subcontexts for triadic data.

Double Boolean algebras are algebras D _ : = ( D ; ⊓ , ⊔ , ¬ , ⌟ , ⊥ , ⊤ ) of type ( 2 , 2 , 1 , 1 , 0 , 0 ) introduced by Rudolf Wille to capture the equational theory of the algebra of protoconcepts. Every double Boolean algebra D _ contains two Boolean algebras: D _ ⊓ and D _ ⊔ . Three main goals are achieved in this paper. First we characterize sub-algebras of a double Boolean algebra D _ as join sets of sub-algebras of the Boolean algebras D _ ⊓ and D _ ⊔ and a subset of D ﹨ D p (where D p = D ⊓ ∪ D ⊔ ) satisfying certain conditions. Second, we characterize homomorphisms between two double Boolean algebras D _ and E _ by homomorphisms between the Boolean algebras D _ ⊓ and E _ ⊓ , D _ ⊔ and E _ ⊔ and maps between D ﹨ D p and E satisfying certain conditions. Third, we give some tools to construct some classes of pure double Boolean algebras.

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 $(\varphi, \psi)$ a Galois connection between $A_1^G$ and $A_2^M$ that is compatible with the residuation, then we show that \[\mathcal{C}: =\{(h,f)\in A_1^G\times A_2^M; \varphi(h)=f \text{ and } \psi(f)=h \}\] can be endowed with a complete residuated multilattice structure. This is a generalization of a result by Ruiz-Calvi{\~n}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. Comment: 21 pages, 6 figures

Dicomplemented lattices were introduced as an abstraction of Wille’s concept algebras which provided negations to a concept lattice. We prove a discrete representation theorem for the class of dicomplemented lattices. The theorem is based on a topology free version of Urquhart’s representation of g eneral lattices.

The search for cohesive groups inside a social network is a topic commonly known as community detection and has attracted many researchers. However, the identification of groups with competitive features using blockmodeling, biclustering and structural or regular equivalences has benefited from a less important interest within the research community. In this paper we define a generic biclustering method called BiP that computes semantically meaningful coclusters (or biclusters) from a two-mode data network. The method has the following features: (i) it allows the processing of adjacency matrices whose data type can be either binary, discrete or categorical without any need for prior data codification, and (ii) each generated block may either represent a cluster of objects with the properties they own (or not), or a juxtaposition of subgroups with distinct profiles (and sometimes purely opposed ones) for a subset of attributes.