Universal Algebra and Model Theory

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This project concerns the distribution and structure of irreducible models of equational theories of algebras. The chief goal is to determine the structural implications for the models of a theory which arise from the assumption that the theory has a bounded number of irreducible models. The projected results have applications to the decidability, finite axiomatizability, categoricity, and equational completeness of equational classes of algebras. Algebraic structures, or algebras, are mathematical objects which are used as devices for calculation. A typical example of an algebra is the number system we use for counting, although more exotic algebras find application in physics, chemistry, logic, and most branches of mathematics. Basic questions in algebra, such as the question of whether one number divides another, are solved by reducing the question to a related question about irreducible factor algebras. This research will advance the understanding of methods of calculation in irreducible algebras, and consequently in general algebras.