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Denote by Id$_c G$ the lattice of all principal $\ell$-ideals of an Abelian $\ell$-group $G$. Our main result is the following. Theorem. For every countable Abelian $\ell$-group $G$, every countable completely normal distributive 0-lattice $L,$ and every closed 0-lattice homomorphism $\varphi : {\rm Id}_c G \to L$, there are a countable Abelian $\ell$-group $H$, an $\ell$-homomorphism $f: G \to H$, and a lattice isomorphism $\iota: {\rm Id}_c H \to L$ such that $\varphi = \iota \circ {\rm Id}_c f$. We record the following consequences of that result: (1) A 0-lattice homomorphism $\varphi: K \to L$, between countable completely normal distributive 0-lattices, can be represented, with respect to the functor Id$_c$, by an $\ell$-homomorphism of Abelian $\ell$-groups iff it is closed. (2) A distributive 0-lattice $D$ of cardinality at most $\aleph_1$ is isomorphic to some Id$_c G$ iff $D$ is completely normal and for all $a,b \in D$ the set $\{x\in D | a \leq b \vee x$ has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to $\aleph_1$. The bound $\aleph_1$ is sharp. Item (1) is extended to commutative diagrams indexed by forests in which ever

Author

Ploscica, Miroslav; Wehrung, Friedrich

Published

2023

Language

EN