Boolean topological graphs of semigroups: the lack of first-order axiomatization |
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Authors: | Micha? M Stronkowski Belinda Trotta |
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Institution: | 1. Faculty of Mathematics and Information Sciences, Warsaw University of Technology, ul. Koszykowa 75, 00-662, Warsaw, Poland 2. AGL Energy, 120 Spencer St, Melbourne, Victoria, 3000, Australia
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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 ?? of algebras let G(??)={G(A)∣A∈??}. Assume that ?? is a class of semigroups possessing a nontrivial member with a neutral element and let ? be the universal Horn class generated by G(??). We prove that the Boolean core of ?, i.e., the topological prevariety generated by finite members of ? 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 ?. We derive analogous results when ?? is a class of monoids or groups with a nontrivial member. |
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