Cyclic congruences of slim semimodular lattices and non-finite axiomatizability of some finite structures

We give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among graphs. (This is consequence general theorem proved L. Ham and M. Jackson, counterpart this for all in class well-known compactness theorem.) Also, to exemplify our method applicable various fields mathematics, we prove neither simple groups, nor ordered sets join-irreducible congruences slim semimodular lattices can described axioms structures. Since 2007 result G. Grätzer E. Knapp, have constituted most intensively studied part lattice theory they already led results even group geometry. In addition non-axiomatizability mentioned above, present property, called Decomposable Cyclic Elements Property, congruence lattices.