Boolean topological graphs of semigroups: the lack of first-order axiomatization

Abstract

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class C of algebras letAssume that C is a class of semigroups possessing a nontrivial member with a neutral element and let H be the universal Horn class generated by G(C ). We prove that the Boolean core of H , i.e., the topological prevariety generated by finite members of H equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of H . We derive analogous results when C is a class of monoids or groups with a nontrivial member.