Model Theory of Epimorphisms

Paul D. Bacsich

doi:10.4153/cmb-1974-083-6

Model Theory of Epimorphisms

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-083-6 ◽

1974 ◽

Vol 17 (4) ◽

pp. 471-477 ◽

Cited By ~ 6

Author(s):

Paul D. Bacsich

Keyword(s):

Model Theory ◽

Order Theory ◽

First Order ◽

First Order Theory

Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .

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The model-theoretic significance of complemented existential formulas

Journal of Symbolic Logic ◽

10.2307/2273232 ◽

1981 ◽

Vol 46 (4) ◽

pp. 843-850 ◽

Cited By ~ 1

Author(s):

Volker Weispfenning

Keyword(s):

Distributive Lattice ◽

Model Theory ◽

Order Theory ◽

Equivalence Classes ◽

First Order ◽

First Order Theory

Let T be an inductive, first-order theory in a language L, let E(L) denote the set of existential L-formulas, and let E(T) denote the distributive lattice of equivalence-classes φT of formulas φ ∈ E(L) with respect to equivalence in T. We consider three types of ‘complements’ in E(T): Let φT, ψT ∈ E(T) and suppose φT ∏ ψT = 0. Then ψT is a complement of φT, if φT ∐ ψT = 1; ψT is a pseudo-complement of φT, if for all μT ∈ E(T), (φT ∐ ψT) = 0 implies μT ≤ ψT; ψT is a “weak complement of φT, if for all μT ∈ E(T), (φT ⋰ ψT) ∐ μT = 0 implies μT = 0. The following facts are obvious: A complement of φT is also a pseudo-complement of φT and a pseudo-complement of φT is also a weak complement of φT. Any φT has at most one pseudo-complement; it is denoted by φT*. The relations ‘ψT is the complement of φT’ and ‘ψT is a weak complement of φT’ are symmetrical. We call φT (weakly, pseudo-) complemented if φT has a (weak, pseudo-) complement, and we call E(T) (weakly, pseudo-) complemented if every φT is (weakly, pseudo-) complemented.The object of this note is to characterize (weakly, pseudo-) complemented existential formulas in model-theoretic terms, and conversely to characterize some classical notions of Robinson style model theory in terms of these formulas. The following theorems illustrate the second approach.

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THE FIRST-ORDER THEORY OF BRANCH GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788715000762 ◽

2016 ◽

Vol 102 (1) ◽

pp. 150-158 ◽

Cited By ~ 1

Author(s):

JOHN S. WILSON

Keyword(s):

Model Theory ◽

Order Theory ◽

Order Model ◽

First Order ◽

First Order Theory

It is shown that for many branch groups $G$ the action on the ambient tree can be interpreted in $G$, in the sense of first-order model theory.

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A new proof of a theorem of Shelah

Journal of Symbolic Logic ◽

10.2307/2272555 ◽

1972 ◽

Vol 37 (1) ◽

pp. 133-134 ◽

Cited By ~ 5

Author(s):

John W. Rosenthal

Keyword(s):

Model Theory ◽

Continuum Hypothesis ◽

Order Theory ◽

Transfinite Induction ◽

Original Proof ◽

First Order ◽

Elementary Chain ◽

First Order Theory ◽

Definable Subset ◽

Generalized Continuum

In [10, §0, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory T is ℵ0-stable (totally transcendental) and not ℵ1-categorical, then it has at least ∣1 + α∣ models of power ℵα.In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of ℵ0-stability and ℵ1-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show of power ℵα such that there is a countable definable subset in . Let B0 be such a subset. Say . We will give by transfinite induction an elementary chain of models of T of power ℵα such that B[i1 … in] has power ℵβ and such that every infinite definable subset of has power ≥ℵβ. This clearly suffices.

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POSITIVE MODEL THEORY AND COMPACT ABSTRACT THEORIES

Journal of Mathematical Logic ◽

10.1142/s0219061303000212 ◽

2003 ◽

Vol 03 (01) ◽

pp. 85-118 ◽

Cited By ~ 33

Author(s):

ITAY BEN-YAACOV

Keyword(s):

Model Theory ◽

Hilbert Spaces ◽

Classical Model ◽

Order Theory ◽

Abstract Theory ◽

First Order ◽

First Order Theory

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

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