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

TL;DR

The paper demonstrates that certain finite structures, including bipartite graphs and slim semimodular lattices, cannot be finitely axiomatized within their respective classes, using a new proof technique applicable across various mathematical fields.

Contribution

It introduces a novel proof method for non-finite axiomatizability results and applies it to finite bipartite graphs, simple groups, and slim semimodular lattices, expanding understanding in model theory and lattice theory.

Findings

Finite bipartite graphs are not finitely axiomatizable among finite graphs.

Finite simple groups cannot be finitely axiomatized in the class of finite structures.

Slim semimodular lattices' congruence lattices are also non-finitely axiomatizable.

Abstract

We give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among FINITE graphs. (This fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and the counterpart of this fact 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. In addition to the non-axiomatizability results mentioned above, we present a new property, called Decomposable Cyclic Elements Property, of the congruence lattices of slim semimodular lattices.