Multi-adjoint property-oriented and object-oriented concept lattices

This paper presents a generalisation of the classical property and object-oriented concept lattices to a fuzzy environment based on the philosophy of the multi-adjoint paradigm. These concept lattices are generalisations of rough set theory used to consider two different sets – the set of objects and the set of attributes – to apply the corresponding modal operators, as in formal concept analysis.First of all, the paper presents several specific properties of adjoint triples in which the duals of several supports are considered. These properties are then used to introduce the multi-adjoint property and object-oriented concept lattices, in which different adjoint triples can be used in non-linear sets, as well as the corresponding representation (fundamental) theorems. Moreover, these lattices are related to the multi-adjoint concept lattice, in which negation operators are not needed; as a result, this relation allows the properties to be translated from one to another.