An adjunction involving l-groups with a strong unit

JOSÉ LUIS CASTIGLIONI; VICENZO MARRA; HERNÁN JAVIER SAN MARTÍN

Buenos Aires

Encuentro; 3rd. MaToMUVI Meeting; 2015

UBA, CONICET, ANCEFN

The correspondence that assigns to each (abelian) l-group A with a strong unit its prime spectrum Spec(A) can be extended to a functor from the category of (abelian) l-groups with a strong unit to the category of root systems with p-morphisms as arrows. Since each categorical equivalence must reflect isomorphisms and Spec(Z,1) = Spec(R,1) although (Z,1) is not isomorphic to (R,1), it follows that Spec may be not part of a categorical equivalence. Hence, it naturally arise the question about the possibility for Spec (or some variation of it) to form part of an adjunction pair. Let us write LGu for the category whose objects are (abelian) l-groups with finite prime spectrum endowed with a family of archimedean elements, one for each prime l-ideal, which satisfy certain relations, and whose arrows are l-group homomorphisms with additional structure. We shall show that a functor which to each object  (A, hat{u}) in LGu assigns the prime spectrum of A, and to each arrow f in LGu assigns a p-morphism, has a left adjoint.