A remark on Martin's Conjecture

2001 ◽  
Vol 66 (1) ◽  
pp. 401-406
Author(s):  
Su Gao
Keyword(s):  
Order Theory ◽  
Boolean Algebras ◽  
First Order ◽  
First Order Theory ◽  
Elementary Theories

AbstractWe prove that the strong Martin conjecture is false. The counterexample is the first-order theory of infinite atomic Boolean algebras. We show that for this class of Boolean algebras, the classification of their (ω + ω)-elementary theories can be reduced to the classification of the elementary theories of their quotient algrbras modulo the Frechét ideals.

On decidable varieties of Heyting algebras

1992 ◽  
Vol 57 (3) ◽  
pp. 988-991 ◽  
Author(s):  
Devdatt P. Dubhashi
Keyword(s):  
Order Theory ◽  
Boolean Algebras ◽  
Semantic Interpretation ◽  
Structure Theory ◽  
Heyting Algebras ◽  
Quantitative Classification ◽  
First Order ◽  
First Order Theory ◽  
Famous Result

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

Automata Presenting Structures: A Survey of the Finite String Case

2008 ◽  
Vol 14 (2) ◽  
pp. 169-209 ◽  
Author(s):  
Sasha Rubin
Keyword(s):  
Order Theory ◽  
Isomorphism Problem ◽  
First Order ◽  
First Order Theory ◽  
Automatic Presentations ◽  
The Relationship

AbstractA structure has a (finite-string)automatic presentationif the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

scholarly journals THE LINDENBAUM ALGEBRA OF THE THEORY OF THE CLASS OF ALL FINITE MODELS

2002 ◽  
Vol 02 (02) ◽  
pp. 145-225 ◽  
Author(s):  
STEFFEN LEMPP ◽  
MIKHAIL PERETYAT'KIN ◽  
REED SOLOMON
Keyword(s):  
Boolean Algebra ◽  
Order Theory ◽  
Boolean Algebras ◽  
Quotient Algebra ◽  
First Order ◽  
Finite Models ◽  
First Order Theory ◽  
Recursive Isomorphism ◽  
Atomic Boolean Algebra

In this paper, we investigate the Lindenbaum algebra ℒ(T fin ) of the theory T fin = Th (M fin ) of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra (ℒ(T fin )/ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: see text] Boolean algebras. These conditions characterize uniquely the algebra ℒ(T fin ); moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra (ℒ(T fin )/ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature.

A proof of morley's conjecture

1989 ◽  
Vol 54 (4) ◽  
pp. 1346-1358
Author(s):  
Bradd Hart
Keyword(s):  
Order Theory ◽  
Exact Calculation ◽  
First Order ◽  
First Order Theory ◽  
Maximal Spectrum ◽  
Uncountable Cardinals ◽  
Naive Approach

In the 1960's, it was conjectured that a complete first order theory in a countable language would have a nondecreasing spectrum on uncountable cardinals. This conjecture became known as Morley's conjecture. Shelah has proved this in [10]. The intent of this paper is to give a different proof which resembles a more naive way of approaching this theorem.Let I(T, λ) = the number of nonisomorphic models of T in cardinality λ. We prove:Theorem 0.1. If T is a complete countable first order theory then for ℵ0 < κ < λ, I(T,K) ≤ I(T, λ).In some sense, one can view Shelah's work on the classification of first order theories as an attack on Morley's conjecture. Over the years, he has shown that certain assumptions on a first order theory would lead to its having maximal spectrum in powers larger than the cardinality of its language (see §6 for precise references). At some point it must have seemed that Morley's conjecture would be a corollary to an exact calculation of all possible spectrums. In the end, this did not occur and, in fact, the exact spectrum functions are still not known (see [10]). Let us consider a naive approach to the proof.If we have two nonisomorphic models of the same cardinality and their cardinality is “large enough” then there should be some reason, irrespective of their cardinalities, which causes this nonisomorphism. If we could isolate this property and extend these models to a larger cardinality preserving this property, then the larger models would also be nonisomorphic. The notion of extendibility introduced in §2 is such a property which allows a version of this naive proof to work. Let us preview the sections.

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory
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