Elementary Equivalence Research Articles

Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Σ-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.

Elementary equivalence for the lattices of subalgebras and automorphism groups of free algebras

We find an elementary equivalence criterion for the lattices of subalgebras of free algebras in regular varieties. The question is addressed of elementary equivalence for the automorphism groups of algebras of this type.

Elementary equivalence of Chevalley groups over fields

It is proved that (elementary) Chevalley groups G π(Φ,K) and G π′(Φ′,K′) (or E π(Φ,K) and E π′(Φ′,K′)) over infinite fields K and K′ of characteristic different from 2, with weight lattices Λ and Λ′, respectively, are elementarily equivalent if and only if the root systems Φ and Φ′ are isomorphic, the fields K and K′ are elementarily equivalent, and the lattices Λ and Λ′ coincide.

Elementary equivalence of some rings of definable functions

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from Rn to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition such as continuity.

How enumeration reductibility yields extended Harrington non-splitting

§1. Introduction. Sacks [16] showed that every computably enumerable (c.e.) degree > 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [11] proved the existence of a c.e. a < 0 which has no c.e. splitting above some proper c.e. predecessor, and Harrington [10] showed that one could take a = 0′. Splitting and nonsplitting techniques have had a number of consequences for definability and elementary equivalence in the degrees below 0′.Heterogeneous splittings are best considered in the context of cupping and non-cupping. Posner and Robinson [15] showed that every nonzero Δ2 degree can be nontrivially cupped to 0′, and Arslanov [1] showed that every c.e. degree > 0 can be d.c.e. cupped to 0′ (and hence since every d.c.e., or even n-c.e., degree has a nonzero c.e. predecessor, every n-c.e. degree > 0 is d.c.e. cuppable). Cooper [4] and Yates (see Miller [13]) showed the existence of degrees noncuppable in the c.e. degrees. Moreover, the search for relative cupping results was drastically limited by Cooper [5], and Slaman and Steel [17] (see also Downey [9]), who showed that there is a nonzero c.e. degree a below which even Δ2 cupping of c.e. degrees fails.We prove below what appears to be the strongest possible of such nonsplitting and noncupping results.

Nondiversity in substructures

AbstractFor a model of Peano Arithmetic, let Lt() be the lattice of its elementary substructures, and let Lt+ () be the equivalenced lattice (Lt(),≅), where ≅ is the equivalence relation of isomorphism on Lt(). It is known that Lt+() is always a reasonable equivalenced lattice.Theorem. Let L be a finite distributive lattice and let (L, E) be reasonable. If 0 is a nonstandard prime model of PA, then 0 has a cofinal extension such that Lt+() ≅ (L,E).A general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices.

Elementary equivalence of the semigroup of invertible matrices with nonnegative elements

In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly ordered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are elementarily equivalent as ordered rings.

Internally club and approachable for larger structures

We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of Pκ(X), where κ ⊆ X. Suppose μ < κ, μ<μ = μ, and κ is supercompact. Then there is a generic extension in which κ = μ++, and for all regular λ ≥ μ++, there are stationarily many N in [H(λ)]μ which are internally club but not internally approachable. Suppose μ is an infinite cardinal. A set N is internally approachable with length μ if N is the union of an increasing and continuous sequence 〈Ni : i < μ〉 of sets with size μ such that for all α < μ, 〈Ni : i < α〉 is in N . A related idea is that of an internally club set. A set N with size μ is internally club if N ∩ [N ] contains a club subset of [N ]. In other words, N is the union of an increasing and continuous sequence 〈Ni : i < μ〉 of sets with size μ such that each Ni is in N . Foreman and Todorcevic [3] asked whether the properties of being internally approachable and internally club are equivalent. In [5] we proved that under PFA, for all regular λ ≥ ω2 there are stationarily many structures N ≺ H(λ) with size א1 such that N is internally club but not internally approachable. In this paper we generalize this result to larger structures. Theorem 1. Suppose μ < κ, μ = μ, and κ is supercompact. Then there is a μ-closed, μ-proper forcing poset which collapses κ to become μ, and forces that for all regular λ ≥ μ, there are stationarily many N in [H(λ)]μ+ which are internally club but not internally approachable. In the model we construct to prove Theorem 1, we have that 2 = μ. In fact, if 2 = μ, then any elementary substructure N ≺ H(λ) with size μ and which contains μ is internally club iff it is internally approachable; this is shown at the end of the paper. In Section 1 we review notation and some background material. Section 2 generalizes the idea of a fat subset of a regular cardinal κ to a fat subset of Pκ(X), where κ ⊆ X. Section 3 presents the basic forcing poset we use in our consistency result, and in Section 4 we describe how to iterate this poset with a mixed support forcing iteration. In Section 5 we prove Theorem 1.

K-analytic versus ccm-analytic sets in nonstandard compact complex manifolds

It is shown that in an elementary extension of a compact complex manifold $M$, the $K$-analytic sets (where $K$ is the algebraic closure of the underlying real closed field) agree with the ccm-analytic sets if and only if $M$ is essentially saturated. In

The McKinsey–Lemmon logic is barely canonical

We study a canonical modal logic introduced by Lemmon, and axiomatised by an infinite sequence of axioms generalising McKinsey’s formula. We prove that the class of all frames for this logic is not closed under elementary equivalence, and so is non-elementary. We also show that any axiomatisation of the logic involves infinitely many non-canonical formulas.

On Extensions of Elementary Submodels by Forcing

We study when a partial order ℙ preserves certain properties of an elementary submodel (of a structure of the form H θ ), including countable closure and ω-covering. In particular, we discuss the existence of ω-covering elementary submodels of different sizes.

Games on Trees and Syntactical Complexity of Formulas

Ehrenfeucht-Fraisse games have been introduced as a means of characterizing the relation of elementary equivalence between structures, or relational database instances in first order logic (FO), or equivalently Relational Calculus. In∼the usual Ehrenfeucht-Fraisse games the rules are determined by a linear ordering of a fixed lenght or, equivalently, by a special kind of tree - a chain of a fixed length -, where each point of that ordering or node of that tree corresponds to a quantification operation. Here we consider Ehrenfeucht-Fraisse games whose rules are determined by arbitrary trees such that their nodes correspond either to quantification operations (“q-nodes”) or to connective operations (“c-nodes”). By playing games on trees, we can refine the class of sentences which are considered in a given game, since a tree represents a particular class of sentences. We define and study several variations of tree games, for first and second order logic (SO). We give a sufficient condition for FO and SO equivalence restricted to formulae with up to n connectives, and hence also a sufficient condition for the non expressibility of a given query in those logics with formulae whose number of logical connectives is less than a given integer. We also give a sufficient condition to prove simultaneous lower bounds in both the number of connectives and in the quantifier types needed to express a given property in FO. If we consider only quantifier types, we get a characterization of the relation of preservation of sentences in the fragment of FO with the given set of quantifier types. We also study tree games for Σ n and Π n formulae. To illustrate the use of our games we use them to prove lower bounds in the connective size for several FO queries, like size of a database, size of a clique in a graph, size of a unary relation, transitive property in a graph, and degree of a node in a graph. Regarding SO, we prove lower bounds for quantifier rank for the parity query.Finally, we give a precise characterization of the logic whose elementary equivalence is characterized by a given tree game, as well as several equivalent characterizations of the existence of a winning strategy for Duplicator in the classical Ehrenfeucht-Fraisse game.

On the density and net weight of regular spaces

We use the cardinal functions ac and lc, due to Fedeli, to establish bounds on the density and net weight of regular spaces which improve some well known bounds. In particular, we use the language of elementary submodels to establish that d(X) � ��(X) ac(X) for every regular space X. This generalizes the following result due to Shapirovskiuo: d(X) � ��(X) c(X) for every regular space X.

Normality and countable paracompactness of hyperspaces of ordinals

For an ordinal α, 2 α denotes the collection of all nonempty closed sets of α with the Vietoris topology and K ( α ) denotes the collection of all nonempty compact sets of α with the subspace topology of 2 α . It is well known that 2 α is normal iff cf α = 1 . In this paper, we will prove that for every nonzero-ordinal α: (1) 2 α is countably paracompact iff cf α ≠ ω . (2) K ( α ) is countably paracompact. (3) K ( α ) is normal iff, if cf α is uncountable, then cf α = α . In (3), we use elementary submodel techniques.

Some questions concerning minimal structures

An infinite first-order structure is minimal if its each definable subset is either finite or co-finite. We formulate three questions concerning order properties of minimal structures which are motivated by Pillay?s Conjecture (stating that a countable first-order structure must have infinitely many countable, pairwise non-isomorphic elementary extensions).

Nonstandard models that are definable in models of Peano Arithmetic

AbstractIn this paper, we investigate definable models of Peano Arithmetic PA in a model of PA. For any definable model N without parameters in a model M, we show that N is isomorphic to M if M is elementary extension of the standard model and N is elementarily equivalent to M. On the other hand, we show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M but N is not isomorphic to M. We also show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M, and N is isomorphic to M, but N is not definably isomorphic to M. And also, we give a generalization of Tennenbaum's theorem. At the end, we give a new method to construct a definable model by a refinement of Kotlarski's method. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Model-theoretic aspects of [formula omitted]-cotorsion modules

Let R be an associative ring with identity. It is shown that every Σ -cotorsion left R -module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ -cotorsion. It follows that, for countable R , the class of Σ -cotorsion modules is closed under elementary equivalence and pure submodules. The modules M that satisfy this descending chain condition are the cotorsion analogues of totally transcendental modules; we characterize them as the modules M for which Ext 1 ( F , M ( ℵ 0 ) ) = 0 , for every countably presented flat module F .

Compact spaces, elementary submodels, and the countable chain condition

Given a space 〈 X , J 〉 in an elementary submodel M of H ( θ ) , define X M to be X ∩ M with the topology generated by { U ∩ M : U ∈ J ∩ M } . It is established, using anti-large-cardinals assumptions, that if X M is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X = X M . Assuming CH + SCH (the Singular Cardinals Hypothesis) in addition, the result holds for any compact X M satisfying the countable chain condition.

A polarized partition relation for weakly compact cardinals using elementary substructures

AbstractWe show that if κ is a weakly compact cardinal, thenfor any ordinals α &lt; κ+ and μ &lt; κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partitionof κ × κ+ into m + μ pieces either there are A ∈ [κ]κ, B ∈ [κ]+]α and i &lt; m with A × B ⊆ Ki or there are C ∈ [κ]κ, , and j &lt; μ with C × D ⊆ Lj. Related results for measurable and almost measurable κ are also investigated. Our proofs of these relations involve the use of elementary substructures of set models of large fragments of ZFC.

Elementary properties of categories of acts over monoids

We explore connections between elementary equivalence of categories of acts over monoids and second-order equivalence of monoids.