Boolean topological graphs of semigroups: the lack of first-order axiomatization
MichaΕ M. Stronkowski, Belinda Trotta Β· Semigroup Forum Β· 2014
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 π of algebras let G (π)={ G ( A )β£ A βπ}. Assume that π is a class of semigroups possessing a nontrivial member with a neutral element and let β be the universal Horn class generated by G (π). We prove that the Boolean core of β, i.e., the topological prevariety generated by finite members of β 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 β. We derive analogous results when π is a class of monoids or groups with a nontrivial member.