Axiomatizability of classes closed under intersection of submodels

CAMPERCHOLI, MIGUEL; VAGGIONE, DIEGO

Buenos Aires

Congreso; Logic Computability and Randomness; 2007

Facultad de Ciencias Exactas y Naturales, UBA

    A class of models C is Closed Under Intersection of Submodels provided that for all A1,A2,B in C    If A1,A2 are in S(B) and the intersection of A1 and A2 not empty, then  intersection of A1 and A2 is in C.    We prove the following:<theorem/>Let A be a finite model of a first order language without one-element submodels. Then every C contained in S(A) which is closed under intersection of submodels is axiomatizable by (A)(E)!-sentences (relative to S(A)) if and only if     (1) no two distinct submodels of A are isomorphic and     (2) for B in S(A) and f,g in Aut(B), if f(b)=g(b) for some b in B then f=g.    Some applications to discriminator varieties will be given.