Cartesian closedness in categories of relational systems (Q1123254)

Two notions, called reflexivity and diagonality, for relational systems (considered as sets of maps from an index set I to some set X) are defined. A system is reflexive if it contains all constant maps. It is diagonal if (for \(I\neq \emptyset)\) the existence of two families of maps \(\{r_ i| i\in I\}\) and \(\{s_ j| j\in I\}\) such that \(s_ i(j)=r_ j(i)\) implies that the map \(t(i)=r_ i(i)\) (i\(\in I)\) belongs to the system. Let \(REL_ I\) be the full subcategory of the category of relational systems with index set I whose objects are reflexive and diagonal relational systems. The main theorem says that \(REL_ I\) is cartesian closed for every set I. Some examples are considered.