Some adjunctions involving the spectrum functor

J.L. CASTIGLIONI

CABA

Simposio; 16th Latin American Symposium on Mathematical Logic; 2014

CONICET

Let $\mathcal{H}_f$ be the category of finite Heyting algebras. The assigment to each $H$ in $\mathcal{H}_f$ of its (finite) poset of prime filters defines a functor$\mathrm{Spec} : \mathcal{H}_f^{op} \to Pos_f$. It is well known that this functor is part of a categorical equivalence (a particular case of Esakia duality). If we restrict$\mathrm{Spec}$ to the full subcategory of $\mathcal{H}_f$ whose objects are the prelinear algebras (finite G\"odel algebras), we get the equivalence$\mathrm{Spec} : \mathcal{G}_f^{op} \to \mathcal{F}_f$, where $\mathcal{F}_f$ is the category of finite forests with p-morphisms as arrows. On the other hand, it is also well known that a (finite) poset is isomorphic to the prime spectrum of a unital abelian $\ell$-group if and only if it isa (finite) forest. It is the case that {$\mathrm{Spec}:(\ell-\mathcal{A}^{u})_{fs}^{op} \to \mathcal{F}_f$} is a functor from the category of unital abelian $\ell$-group with finite spectrum tothe category of finite forests. It is hence natural to ask whether this functor is part of a categorical equivalence or not; and if not, if it has right or left adjoints. In this talk we shall answer this question and state some related ones.