Tensor products of contexts and complete lattices
Contexts, which play a fundamental role in \textit{R. Wille}'s formal concept analysis [in: Ordered sets, Proc. NATO Adv. Stud. Inst., Banff/Can. 1981, 445-470 (1982; Zbl 0491.06008)], are just binary relations between sets. Although they agree with Boolean Chu spaces as recently studied by Pratt and others, the notion of conceptual morphism between contexts is fundamentally different from the notion of a Chu transform, so the analogy ends right here. The concept lattice of a context essentially is the set of fixed points of the polarity (= Galois connection between the power sets) induced by the relation, suitably ordered. Consequently, the concept lattice of the order relation of a poset is essentially the Dedekind-MacNeille completion of the poset. Conversely, the order relation of a concept lattice may ve viewed as a completion of the original context. This induces a reflection on \textbf{CC}, and the corresponding reflective subcategory may be identified with the category \textbf{CLC} of complete lattices and complete homomorphisms. \textit{R. Wille} introduced a binary operation called tensor product for complete lattices [Order 2, 81-95 (1985; Zbl 0583.06007)], which he later generalized to arbitrarily many factors. Instead of the pointwise order on the cartesian product of the underlying sets, one considers the relation between tuples that requires the elements in at least one component to satisfy the order relation of the corresponding factor, and then one passes to the concept lattice. Besides generalizing this notion from \textbf{CLC} to \textbf{CC}, for the original construction in \textbf{CLC} the author constructs a canonical family of complete homomorphisms from the factors into the tensor product. He then shows that this family has a universal property in \textbf{CLC} just like that for coproducts, however not with respect to arbitrary sinks (= families of complete homomorphisms with common codomain), but with respect to so-called distributing sinks. Since every sink with a completely distributive codomain is distributing, and the tensor product of completely distributive factors is again completely distributive, on the corresponding full subcategory \textbf{CDC} of \textbf{CLC} the tensor product in fact coincides with the categorical coproduct.
scientific article; zbMATH DE number 513019
scientific article