On clones of relations (Q1177861)

The category \(\underline{\hbox{Par}}\) of all partial functions between arbitrary sets forms a diagonal halfterminal symmetric category with diagonal inversion. Since \(n\)-ary relations are definable as partial projections in such a category, the usual relation operations like cyclical permutation of components, exchange of first and second component, addition of a fictive component, identification of two components, intersection of two relations, cartesian product of two relations, and a modified composition of relations arise in a natural manner from the structure of a \(\hbox{dht} \triangledown\)-symmetric category. Taking into consideration two additional conditions which are fulfilled in \(\underline{\hbox{Par}}\), and therefore in each partial theory, one gets also the cancellation of components and the usual relation folding. Each so-called simple relation is derivable from the special simple relation \(\delta=(1_ A\otimes \triangledown_ A)p^ 2_ 1\leq p^ 3_ 1\) \(((x_ 1,x_ 2,x_ 3)\in \delta\) iff \(x_ 2=x_ 3\)). In the language of \(\hbox{dht}\)-symmetric theories it was shown that the set \(\hbox{Pol} Q\) of all polymorphisms of all relations \(q\in Q\) is the carrier of a function clone. The set \(\hbox{Inv} F\) of all invariants of a set \(F\) of functions forms a relation clone. The proof of this result requires the language of partial theories, i.e. the additional conditions as mentioned above.