The relational systemRelational systems change a regular basis. They add new data types, new output functions, andnew relationship capabilities. They have always been downward compatible, thus preserving theability to work with older data.Given a set E, a relational system with base E, or based on E, is any ordinal sequence whoseterms are relations Ri (i ordinal) based on E. Letting ni denote the arity of system, the ordinalsequence of integers ni is said to be the arity of the system, and each Ri is said to be a componentof the system.By taking a finite sequence of relations based on E, we find again the multirelationbased on E.The notions of restriction of a system to a subset of the base, extension to a superset of the base,isomorphism, automorphism, local isomorphism or automorphism, all extend immediately to thecase of relational systems.However we have an important difference which prohibits certain generalizations. Indeed thenumber of relational systems of a given arity and of a given finite base is in general infinite. Inparticular RAMSEY's theorem, used when partitioning the p-element subsets of the base into afinite number of classes (colors) corresponding to different isomorphism types, can no longer beused systematically. The same remark holds for the coherence lemma.A relational system R is said to be p-homogeneous iff every local automorphism of R, defined onp elements, is extendible to an automorphism of R.A relational system is said to be homogeneous iff it is p-homogeneous for every integer p. Thesedefinitions generalize those of 12.1.1.Relational Calculus