Real spectra and ℓ-spectra of algebras and vector lattices over countable fields

Wehrung, Friedrich

doi:10.1016/j.jpaa.2021.106861

Cited by 5 publications

(1 citation statement)

References 29 publications

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“…In [41] it is shown that second countable spectral spaces which are prime spectra of MV-algebras are characterized by a simple first order formula, whereas in [42] is is shown that no infinitary first order logic can express the prime spectra of MV-algebras. Other information is contained in [43].…”

Section: Introductionmentioning

confidence: 99%

The spectrum problem for $\ell$-groups and for MV-algebras: a categorical approach

Barbieri¹,

Nola²,

Lenzi³

2023

Preprint

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As a main result, we characterize prime spectra of abelian lattice ordered groups. Further we introduce some categories based on spectral spaces, lattices and Priestley spaces, and we relate these categories with each other and with the category of presented MV-algebras, by means of functors. We turn to lattices and offer a simple characterization of 1) maps whose Stone dual preserves closed sets, and 2) closed epimorphisms between distributive lattices as well as their Stone duals. We have a characterization of the variety generated by the Chang MV-algebra and we study this variety. Next we generalize the results to every variety generated by a Komori chain. Finally we discuss homogeneous polynomials in MV-algebras.

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Section: Introductionmentioning

confidence: 99%

The spectrum problem for $\ell$-groups and for MV-algebras: a categorical approach

Barbieri¹,

Nola²,

Lenzi³

2023

Preprint

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Elementary divisors, Hochster duality, and spectra

Rump

2024

Arch. Math.

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Monotone-Cevian and Finitely Separable Lattices

Ploščica,

Wehrung

2024

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A distributive lattice with zero is completely normal if its prime ideals form a root system under set inclusion. Every such lattice admits a binary operation $$(x,y)\mapsto x\mathbin {\smallsetminus }y$$ ( x , y ) ↦ x \ y satisfying the rules $$x\le y\vee (x\mathbin {\smallsetminus }y)$$ x ≤ y ∨ ( x \ y ) and $$(x\mathbin {\smallsetminus }y)\wedge (y\mathbin {\smallsetminus }x)=0$$ ( x \ y ) ∧ ( y \ x ) = 0 — in short a deviation. In this paper we study the following additional properties of deviations: monotone (i.e., isotone in x and antitone in y) and Cevian (i.e., $$x\mathbin {\smallsetminus }z\le (x\mathbin {\smallsetminus }y)\vee (y\mathbin {\smallsetminus }z)$$ x \ z ≤ ( x \ y ) ∨ ( y \ z ) ). We relate those matters to finite separability as defined by Freese and Nation. We prove that every finitely separable completely normal lattice has a monotone deviation. We pay special attention to lattices of principal $$\ell $$ ℓ -ideals of Abelian $$\ell $$ ℓ -groups (which are always completely normal). We prove that for free Abelian $$\ell $$ ℓ -groups (and also free $$\Bbbk $$ k -vector lattices) those lattices admit monotone Cevian deviations. On the other hand, we construct an Archimedean $$\ell $$ ℓ -group with strong unit, of cardinality $$\aleph _1$$ ℵ 1 , whose principal $$\ell $$ ℓ -ideal lattice does not have a monotone deviation.

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Real spectra and ℓ-spectra of algebras and vector lattices over countable fields

Cited by 5 publications

References 29 publications

The spectrum problem for $\ell$-groups and for MV-algebras: a categorical approach

The spectrum problem for $\ell$-groups and for MV-algebras: a categorical approach

Elementary divisors, Hochster duality, and spectra

Monotone-Cevian and Finitely Separable Lattices

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