Non-finite axiomatizability of some finite structures

Gábor Czédli

Published 2021-01-31Version 1

We prove that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences in the class of finite graphs. (The counterpart of this statement for all bipartite graphs in the class of all graphs is a well-known consequence of the compactness theorem.) Also, to exemplify that our method is applicable in various fields of mathematics, we prove that neither finite simple groups, nor the ordered sets of join-irreducible congruences of slim semimodular lattices can be described by finitely many axioms in the class of finite structures. Since a 2007 result of G. Gr\"atzer and E. Knapp, slim semimodular lattices have constituted the most intensively studied part of lattice theory and they have already led to results even in group theory and geometry.