Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one
نویسندگان
چکیده
It is well known that the lattice $${{\,\mathrm{Id_c}\,}}{G}$$ of all principal $$\ell $$ -ideals any Abelian -group G a completely normal distributive 0-lattice; yet not every 0-lattice homomorphic image some , via counterexample cardinality $$\aleph _2$$ . We prove with at most _1$$ elements By Stone duality, this means generalized spectral space compact open sets homeomorphic to subspace -spectrum -group.
منابع مشابه
Free abelian lattice-ordered groups
Let n be a positive integer and FA`(n) be the free abelian latticeordered group on n generators. We prove that FA`(m) and FA`(n) do not satisfy the same first-order sentences in the language L={+,−, 0,∧,∨} if m 6= n. We also show that Th(FA`(n)) is decidable iff n ∈ {1, 2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices. A. M. S. C...
متن کاملSpectra of Abelian Weakly Associative Lattice Groups
The notion of a weakly associative lattice group is a generalization of that of a lattice ordered group in which the identities of associativity of the lattice operations join and meet are replaced by the identities of weak associativity. In the paper, the spectral topologies on the sets of straightening ideals (and on some of their subsets) of abelian weakly associative lattice groups are intr...
متن کاملFinitely Presented Abelian Lattice-Ordered Groups
We give necessary and sufficient conditions for the first-order theory of a finitely presented abelian lattice-ordered group to be decidable. We also show that if the number of generators is at most 3, then elementary equivalence implies isomorphism. We deduce from our methods that the theory of the free MV -algebra on at least 2 generators is undecidable.
متن کاملExistentially Complete Abelian Lattice-ordered Groups
The theory of abelian totally ordered groups has a model completion. We show that the theory of abelian lattice-ordered groups has no model companion. Indeed, the Archimedean property can be captured by a first order V3V sentence for existentially complete abelian lattice-ordered groups, and distinguishes between finitely generic abelian lattice-ordered groups and infinitely generic ones. We th...
متن کاملSpectra of finitely presented lattice-ordered Abelian groups and MV-algebras, part 1
(See e.g. [6] for background.) When X is equipped with a distinguished basis D for its topology, closed under finite meets and joins, one can investigate situations where D is also closed under the implication (2), i.e., where D is a Heyting subalgebra of O (X). Recall that X is a spectral space if it is compact and T0, its collection D of compact open subsets forms a basis which is closed unde...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Acta Scientiarum Mathematicarum
سال: 2023
ISSN: ['0324-5462', '2064-8316', '0001-6969']
DOI: https://doi.org/10.1007/s44146-023-00080-z