The Correspondence Between Monotonic Many Sorted Signatures and Well - Founded Graphs . Part II 1

The graph induced by a many sorted signature is defined as follows: the vertices are the symbols of sorts, and if a sort s is an argument of an operation with result sort t, then a directed edge [s, t] is in the graph. The key lemma states relationship between the depth of elements of a free many sorted algebra over a signature and the length of directed chains in the graph induced by the signature. Then we prove that a monotonic many sorted signature (every finitely-generated algebra over it is locally-finite) induces a well-founded graph. The converse holds with an additional assumption that the signature is finitely operated, i.e. there is only a finite number of operations with the given result sort.