The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces

1974 ◽  
Vol 39 (4) ◽  
pp. 717-731 ◽  
Author(s):  
C. Ward Henson
Keyword(s):  
Nonstandard Analysis ◽  
Cardinal Number ◽  
Elementary Embedding ◽  
First Order ◽  
Saturation Properties ◽  
New Family ◽  
Theoretical Structure ◽  
Order Language ◽  
Language L ◽  
Elementary Topology

The basic setting of nonstandard analysis consists of a set-theoretical structure together with a map * from into another structure * of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make * into an enlargement of [13]. The structures and * may be type-hierarchies as in [11] and [13] or they may be cumulative structures with ω levels as in [14]. The assumption that * is an enlargement of has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that * has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11].This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number κ, * satisfies the κ-isomorphism property (as an enlargement of ) if the following condition holds:For each first order language L with fewer than κ nonlogical symbols, if and are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to * and ), then and are isomorphic.

On order-types of models

1972 ◽  
Vol 37 (1) ◽  
pp. 69-70 ◽  
Author(s):  
Wilfrid Hodges
Keyword(s):  
Order Type ◽  
Linear Ordering ◽  
Infinite Field ◽  
Strong Limit ◽  
First Order ◽  
Order Types ◽  
Order Language ◽  
Language L ◽  
New Feature

Let T be a theory in a first-order language L. Let L have a predicate ν0 ≺ ν1 such that in every model of T, the interpretation of ≺ is a linear ordering with infinite field. The order-type of this ordering will be called the order-type of the model .Several recent theorems have the following form: if T has a model of order-type ξ then T has a model of order-type ζ (see [1]). We shall add one to the list. The new feature of our result is that the order-type ζ may be in a sense “opposite” to ξ. Silver's Theorem 2.24 of [3] is a corollary of Theorem 1 below.Theorem 1. Let κ be a strong limit number (i.e. μ < κ implies 2μ < κ). Suppose λ < κ, and suppose that for every cardinal μ < κ, T has a model with where the order-type of contains no descending well-ordered sequences of length λ. Then for every cardinal μ ≥ the cardinality ∣L∣ of the language L, T has models and such that(a) the field of is the union of ≤ ∣L∣ well-ordered (inversely well-ordered) parts;(b) .The proof is by Ehrenfeucht-Mostowski models; we presuppose [2].

Topological Model Set Deformations in Logic Programming

1989 ◽  
Vol 12 (3) ◽  
pp. 357-399
Author(s):  
Aida Batarekh ◽  
V.S. Subrahmanian
Keyword(s):  
Logic Programming ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Logic Programs ◽  
First Order ◽  
Order Language ◽  
Language L ◽  
General Logic ◽  
Model Set ◽  
Necessary And Sufficient

Given a first order language L, and a notion of a logic L w.r.t. L, we investigate the topological properties of the space of L-structures for L. We show that under a topology called the query topology which arises naturally in logic programming, the space of L-models (where L is a decent logic) of any sentence (set of clauses) in L may be regarded as a (closed, compact) T4-space. We then investigate the properties of maps from structures to structures. Our results allow us to apply various well-known results on the fixed-points of operators on topological spaces to the semantics of logic programming – in particular, we are able to derive necessary and sufficient topological conditions for the completion of covered general logic programs to be consistent. Moreover, we derive sufficient conditions guaranteeing the consistency of program completions, and for logic programs to be determinate. We also apply our results to characterize consistency of the unions of program completions.

Skolem functions and elementary embeddings

1977 ◽  
Vol 42 (1) ◽  
pp. 94-98 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Elementary Embedding ◽  
Elementary Substructure ◽  
First Order ◽  
Skolem Function ◽  
Free Variables ◽  
Skolem Functions ◽  
Order Language ◽  
Special Cases ◽  
Elementary Embeddings ◽  
Positive Results

Let L be an elementary first order language. Let be an L-structure, and let φ be an L-formula with free variables u1, …, un, and υ. A Skolem function for φ on is an n-ary operation f on such that for all . If is an elementary substructure of , then an n-ary operation f on is said to preserve the elementary embedding of into if f(x)∈ for all x ∈ n, and (, f ∣n) ≺ (, f). Keisler asked the following question:Problem 1. If and are L-structures such that ≺ , and if φ (u, υ) is an L-formula (with appropriate free variables), must there be a Skolem function for φ on which preserves the elementary embedding?Payne [6] gave a counterexample in which the language L is uncountable. In [3], [5], the author announced the existence of an example in which L is countable but the structures and are uncountable. The construction of the example will be given in this paper. Keisler's problem is still open in case both the language and the structures are required to be countable. Positive results for some special cases are given in [4].The following variant of Keisler's question was brought to the author's attention by Peter Winkler:Problem 2. If L is a countable language, a countable L-structure, and φ(u, υ) an L-formula, must there be a Skolem function f for φ on such that for every countable elementary extension of , there is an extension of f which preserves the elementary embedding of into ?

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

scholarly journals ON PRODUCTS OF ELEMENTARILY INDIVISIBLE STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 951-971
Author(s):  
NADAV MEIR
Keyword(s):  
New Products ◽  
Elementary Substructure ◽  
First Order ◽  
Order Language ◽  
Image Position

AbstractWe say a structure ${\cal M}$ in a first-order language ${\cal L}$ is indivisible if for every coloring of its universe in two colors, there is a monochromatic substructure ${\cal M}\prime \subseteq {\cal M}$ such that ${\cal M}\prime \cong {\cal M}$. Additionally, we say that ${\cal M}$ is symmetrically indivisible if ${\cal M}\prime$ can be chosen to be symmetrically embedded in ${\cal M}$ (that is, every automorphism of ${\cal M}\prime$ can be extended to an automorphism of ${\cal M}$). Similarly, we say that ${\cal M}$ is elementarily indivisible if ${\cal M}\prime$ can be chosen to be an elementary substructure. We define new products of structures in a relational language. We use these products to give recipes for construction of elementarily indivisible structures which are not transitive and elementarily indivisible structures which are not symmetrically indivisible, answering two questions presented by A. Hasson, M. Kojman, and A. Onshuus.

scholarly journals Word order in Latin locative constructions: a corpus study in Caesar’s De bello gallico

2011 ◽  
Vol 64 (2) ◽  
Author(s):  
Stavros Skopeteas
Keyword(s):  
Word Order ◽  
Information Structure ◽  
Motion Verbs ◽  
Structural Domains ◽  
Corpus Study ◽  
First Order ◽  
Order Language ◽  
Free Word

AbstractClassical Latin is a free word order language, i.e., the order of the constituents is determined by information structure rather than by syntactic rules. This article presents a corpus study on the word order of locative constructions and shows that the choice between a Theme-first and a Locative-first order is influenced by the discourse status of the referents. Furthermore, the corpus findings reveal a striking impact of the syntactic construction: complements of motion verbs do not have the same ordering preferences with complements of static verbs and adjuncts. This finding supports the view that the influence of discourse status on word order is indirect, i.e., it is mediated by information structural domains.

On Axiomatizability of Non-Commutative Lp-Spaces

2007 ◽  
Vol 50 (4) ◽  
pp. 519-534
Author(s):  
C. Ward Henson ◽  
Yves Raynaud ◽  
Andrew Rizzo
Keyword(s):  
Hilbert Spaces ◽  
Normed Space ◽  
The Other ◽  
Space Structures ◽  
Operator Spaces ◽  
Infinite Dimensions ◽  
Schatten Classes ◽  
Lp Spaces ◽  
First Order ◽  
Order Language

AbstractIt is shown that Schatten p-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative Lp-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative Lp spaces. As a consequence, the class of non-commutative Lp-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative Lp-spaces. For p = 1 this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

Characterizations of Axiomatic Categories of Models Canonically Isomorphic to (Quasi-)Varieties

1988 ◽  
Vol 31 (3) ◽  
pp. 287-300 ◽  
Author(s):  
Michel Hébert
Keyword(s):  
Forgetful Functor ◽  
Canonical Isomorphism ◽  
First Order ◽  
Order Language ◽  
Algebraic Language ◽  
Algebraic Constructions

AbstractLet be the category of all homomorphisms (i.e. functions preserving satisfaction of atomic formulas) between models of a set of sentences T in a finitary first-order language L. Functors between two such categories are said to be canonical if they commute with the forgetful functors. The following properties are characterized syntactically and also in terms of closure of for some algebraic constructions (involving products, equalizers, factorizations and kernel pairs): There is a canonical isomorphism from to a variety (resp. quasivariety) in a finitary expansion of L which assigns to a model its (unique) expansion. This solves a problem of H. Volger.In the case of a purely algebraic language, the properties are equivalent to:“ is canonically isomorphic to a finitary variety (resp. quasivariety)” and, for the variety case, to “the forgetful functor of is monadic (tripleable)”.

New family of parasupercoherent states: entanglement, uncertainties, and statistical properties

2020 ◽  
Vol 98 (10) ◽  
pp. 953-958
Author(s):  
Amin Motamedinasab ◽  
Azam Anbaraki ◽  
Davood Afshar ◽  
Mojtaba Jafarpour
Keyword(s):  
Harmonic Oscillator ◽  
Arbitrary Order ◽  
Annihilation Operator ◽  
The Other ◽  
Mandel Parameter ◽  
First Order ◽  
New Family ◽  
Wide Range ◽  
New States ◽  
Minimum Uncertainty

The general parasupersymmetric annihilation operator of arbitrary order does not reduce to the Kornbluth–Zypman general supersymmetric annihilation operator for the first order. In this paper, we introduce an annihilation operator for a parasupersymmetric harmonic oscillator that in the first order matches with the Kornblouth–Zypman results. Then, using the latter operator, we obtain the parasupercoherent states and calculate their entanglement, uncertainties, and statistics. We observe that these states are entangled for any arbitrary order of parasupersymmetry and their entanglement goes to zero for the large values of the coherency parameter. In addition, we find that the maximum of the entanglement of parasupercoherent states is a decreasing function of the parasupersymmetry order. Moreover, these states are minimum uncertainty states for large and also small values of the coherency parameter. Furthermore, these states show squeezing in one of the quadrature operators for a wide range of the coherency parameter, while no squeezing in the other quadrature operator is observed at all. In addition, using the Mandel parameter, we find that the statistics of these new states are subPoissonian for small values of the coherency parameter.

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