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

@inproceedings{Ploica2022SpectralSO, title={Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one}, author={Miroslav Plo{\vs}{\vc}ica and Friedrich Wehrung}, year={2022} }

. It is well known that the lattice Id c 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 Id c 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 Id c G . By Stone duality, this means that every completely normal generalized spectral space…

## Figures from this paper

## References

SHOWING 1-10 OF 19 REFERENCES

### Cevian properties in ideal lattices of Abelian ℓ-groups

- MathematicsForum Mathematicum
- 2021

Abstract We consider the problem of describing the lattices of compact ℓ{\ell}-ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered…

### Spectral spaces of countable Abelian lattice-ordered groups

- MathematicsTransactions of the American Mathematical Society
- 2018

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X,…

### Free vector lattices

- Mathematics
- 1973

An investigation into the algebraic properties of free objects in the category of vector lattices is carried out. It is shown that each ideal of a free vector lattice is a cardinal (direct) sum of…

### Real Spectrum Versus ℓ-Spectrum via Brumfiel Spectrum

- MathematicsAlgebras and Representation Theory
- 2021

It is well known that the real spectrum of any commutative unital ring, and the l-spectrum of any Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We prove…

### The essential cover and the absolute cover of a schematic space

- Mathematics
- 2009

A theorem of Gleason states that every compact space admits a projective cover. More generally, in the category of topological spaces with continuous maps, covers exist with respect to the full…

### From noncommutative diagrams to anti-elementary classes

- MathematicsJ. Math. Log.
- 2021

It is proved that many naturally defined classes are anti-elementary, including the class of all lattices of finitely generated convex l-subgroups of members of any class of l- groups containing all Archimedean l-groups.

### From Objects to Diagrams for Ranges of Functors

- Mathematics
- 2011

Let A, B, S be categories, let F:A-->S and G:B-->S be functors. We assume that for ''many'' objects a in A, there exists an object b in B such that F(a) is isomorphic to G(b). We establish a general…

### Advanced Łukasiewicz calculus and MV-algebras

- Biology, Mathematics
- 2011

This book discusses the construction of free products of MV-algebras, the spectral and the maximal spectral space, and a first-order Lukasiewicz logic with [0, 1]-identity.