MONOMORPHIC RELATIONAL SYSTEMS by DAVID CLARK and PETER KRAUSS in New Paltz, New York (U.S.A.)') I n this study we continue the investigation of KRAUSS [5] and clarify the connection of his results to the closely related work of FRAYSSE [2], FRASNAY [3] and JEAN [a].
A set of sentences of the first order predicate calculus with identity and finitely many predicates is called significantly consistent if it has an infinite model. A universal theory is called universally complete if it is maximally significantly consistent. KRAUSS [5] showed that the models of universally complete universal theories are exactly those relational systems which FRAISSE [2] calls chainable. Using results of FRASNAY [3], JEAN [a] established that universally complete universal theories are finitely axiomatizable. Their proofs are considerably involved and are essentially based on a purely group theoretic result, FRASNAY'S Th&or&me de Recollement. KRAUSS [5] directly constructed such axiom systems for the languages with one, two or three-place predicates. Generalizing his method we show that the Th&or&me de Recollement is equivalent to the finite axiomatizability of universally complete universal theories. Chainable relational systems are mono- morphic, that is, all finite subsystems of the same cardinality are isomorphic. We exhibit a large number of monomorphic relational systems which are not chainable, and give a simple proof of FRASNAY'S result that there are only finitely many isomorphism types of such systems. Although this study is self contained, the reader may find it useful to confer KRAUSS [5] for more detailed information concerning the notions discussed in this paper. l) The main reeult of this paper was announced in CLARK and KRAUSS [ l ] .