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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/219933
Relational methods in algebraic logic
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[eng] This thesis is concerned with three instances of relational methods in algebraic logic.
First, determining which partially ordered sets are isomorphic to the spectrum of a Heyting algebra. This is an open question related to the classical problem of representing partially ordered sets as spectra of bounded distributive lattices or, equivalently, commutative rings with unit. We prove that a root system (the order dual of a forest) is isomorphic to the spectrum of a Heyting algebra if and only if it satisfies a simple order theoretic condition, known as "having enough gaps", and each of its nonempty chains has an infimum. This strengthens Lewis' characterisation of the root systems which are spectra of commutative rings with unit. While a similar characterisation for arbitrary forests currently seems out of reach, we show that a well-ordered forest is isomorphic to the spectrum of a Heyting algebra if and only if it has enough gaps and each of its nonempty chains has a supremum.
Second, Sahlqvist theorem provides sufficient syntactic conditions for a normal modal logic to be complete with respect to an elementary class of Kripke frames. We extend Sahlqvist theory to the fragments of the intuitionistic propositional calculus that include the conjunction connective. This allows us to introduce a Sahlqvist theory of intuitionistic character amenable to arbitrary protoalgebraic deductive systems. As an application, we obtain a Sahlqvist theorem for the fragments of the intuitionistic propositional calculus that include the implication connective and for the extensions of the intuitionistic linear logic.
Third, Blok's celebrated dichotomy theorem proves that each normal modal logic shares its Kripke frames with exactly one or continuurn-many logics. It is an outstanding open problem to characterise the number of logics having the same posets of an axiomatic extension of the intuitionistic propositional calculus. We solve this question in the case of implicative logics, the axiomatic extensions of the implicative fragment of the propositional intuitionistic logic. In this case, a trichotomy holds: every irnplicative logics shares its posets exactly with 1, N0, or 2(No) many logics.
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FORNASIERE, Damiano. Relational methods in algebraic logic. [consulta: 30 de novembre de 2025]. [Disponible a: https://hdl.handle.net/2445/219933]