Congruence lattices of free lattices in non-distributive varieties

We prove that for any free lattice F with at least $\aleph\_2$ generators in any non-distributive variety of lattices, there exists no sectionally complemented lattice L with congruence lattice isomorphic to the one of F. This solves a question formulated by Gr\"{a}tzer and Schmidt in 1962. This yields in turn further examples of simply constructed distributive semilattices that are not isomorphic to the semilattice of finitely generated two-sided ideals in any von Neumann regular ring

A solution to the MV-spectrum Problem in size aleph one

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 every node has countable height.All our results are stated in terms of vector lattices over any countable totally ordered division ring

Comment.Math.Univ.Carolinae 33,3 (1992)541{550 541

A duality for isotropic median algebra

Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one

It is well known that the lattice Idc G of all principal ℓ-ideals of any Abelian ℓ-group G is a completely normal distributive 0-lattice, and that not every completely normal distributive 0-lattice is a homomorphic image of some Idc G, via a counterexample of cardinality ℵ 2. We prove that every completely normal distributive 0-lattice with at most ℵ 1 elements is a homomorphic image of some Idc G. By Stone duality, this means that every completely normal generalized spectral space, with at most ℵ 1 compact open sets, is homeomorphic to a spectral subspace of the ℓ-spectrum of some Abelian ℓ-group

Congruence lattices of free lattices in non-distributive varieties

We prove that for any free lattice F with at least $\aleph_2$ generators in any non-distributive variety of lattices, there exists no sectionally complemented lattice L with congruence lattice isomorphic to the one of F. This solves a question formulated by Grätzer and Schmidt in 1962. This yields in turn further examples of simply constructed distributive semilattices that are not isomorphic to the semilattice of finitely generated two-sided ideals in any von Neumann regular ring