Syntactical Reasons

CAMPERCHOLI, MIGUEL

Bahia Blanca

Congreso; XIV Congreso Dr. Antonio Monteiro; 2017

UNS

It is a long tradition in Universal Algebra and Model Theory to look forsyntactical objetcs underlying structural behaviour. Birkho?´s celebrated HSP-Theorem is a premier example, and arguably the result that started it all. Itsays that a class V of algebraic structures is closed under the the formationof direct products, subalgebras and homomorphic images if and only if it canbe axiomatized by identities. The utility and power of this result (witnessedby its countless applications) comes from the fact that it allows us to replacean abstract set of closure conditions for a concrete object (the set of identites), which can be formally manipulated and made part of our reasonings.Following Birkho?´s seminal theorem many other discoveries explaining se-mantical properties through the existence of syntactical objects have been made(e.g., Malcev conditions, Congruence formulas, etc.). The belief that this kindof results provide both a deeper understanding of structural properties, and thepotential of interesting applications, has been the drive behind the lines of work pursued by our research group at FaMAF in Córdoba. Led by Professor Vag-gione, we have worked on several problems of the aforementioned nature. Inour talk we will survey the results we have obtained and explain the methodsand tools employed. We shall also discuss ongoing and future work.