Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Spectral spaces of countable Abelian lattice-ordered groups


Author: Friedrich Wehrung
Journal: Trans. Amer. Math. Soc. 371 (2019), 2133-2158
MSC (2010): Primary 06D05, 06D20, 06D35, 06D50, 06F20, 46A55, 52A05, 52C35
DOI: https://doi.org/10.1090/tran/7596
Published electronically: October 23, 2018
MathSciNet review: 3894048
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that the $ \ell $-spectrum of an Abelian $ \ell $-group, defined as the set of all its prime $ \ell $-ideals with the hull-kernel topology, is a completely normal generalized spectral space. We establish the following converse of this result.


Theorem. Every second countable, completely normal generalized spectral space is homeomorphic to the $ \ell $-spectrum of some Abelian $ \ell $-group.


We obtain this result by proving that a countable distributive lattice $ D$ with zero is isomorphic to the Stone dual of some $ \ell $-spectrum (we say that $ D$ is $ \ell $-representable) iff for all $ a,b\in D$ there are $ x,y\in D$ such that $ a\vee b=a\vee y=b\vee x$ and $ x\wedge y=0$. On the other hand, we construct a non-$ \ell $-representable bounded distributive lattice, of cardinality $ \aleph _1$, with an $ \ell $-representable countable $ \mathbin {\mathscr {L}}_{\infty ,\omega }$-elementary sublattice. In particular, there is no characterization, of the class of all $ \ell $-representable distributive lattices, by any class of $ \mathbin {\mathscr {L}}_{\infty ,\omega }$ sentences.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 06D05, 06D20, 06D35, 06D50, 06F20, 46A55, 52A05, 52C35

Retrieve articles in all journals with MSC (2010): 06D05, 06D20, 06D35, 06D50, 06F20, 46A55, 52A05, 52C35


Additional Information

Friedrich Wehrung
Affiliation: LMNO, CNRS UMR 6139, Département de Mathématiques, Université de Caen Normandie, 14032 Caen Cedex, France
Email: friedrich.wehrung01@unicaen.fr

DOI: https://doi.org/10.1090/tran/7596
Keywords: Lattice-ordered, abelian, group, ideal, prime, spectrum, representable, spectral space, sober, completely normal, countable, distributive, lattice, join-irreducible, Heyting algebra, closed map, consonance, difference operation, hyperplane, open, half-space
Received by editor(s): April 6, 2017
Received by editor(s) in revised form: September 26, 2017
Published electronically: October 23, 2018
Article copyright: © Copyright 2018 American Mathematical Society