The elementary class of products of totally ordered abelian group

A basic goal in model-theoretic algebra is to obtain the classification of the complete extensions of a given (first-order) algebraic theory.Results of this type, for the theory of totally ordered abelian groups, were obtained first by A. Robinson and E. Zakon [5] in 1960, later extended by Yu. Gurevich [4] in 1964, and further clarified by P. Schmitt in [6].Within this circle of ideas, we give in this paper an axiomatization of the first-order theory of the class of all direct products of totally ordered abelian groups, construed as lattice-ordered groups (l-groups)—see the theorem below. We think of this result as constituing a first step—undoubtedly only a small one—towards the more general goal of classifying the first-order theory of abelian l-groups.We write groups for abelian l-groups construed as structures in the language 〈 ∨, ∧, +, −, 0〉 (“−” is an unary operation). For unproved statements and unexplicated definitions, the reader is referred to [1].

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First order theory of cyclically ordered groups

Annals of Pure and Applied Logic ◽

10.1016/j.apal.2018.04.007 ◽

2018 ◽

Vol 169 (9) ◽

pp. 896-927 ◽

Cited By ~ 3

Author(s):

M. Giraudet ◽

G. Leloup ◽

F. Lucas

Keyword(s):

Order Theory ◽

First Order ◽

First Order Theory ◽

Ordered Groups

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A Characterization of Strongly Dependent Ordered Abelian Groups

Revista Colombiana de Matemáticas ◽

10.15446/recolma.v52n2.77154 ◽

2018 ◽

Vol 52 (2) ◽

pp. 139-159 ◽

Cited By ~ 1

Author(s):

Alfred Dolich ◽

John Goodrick

Keyword(s):

Abelian Groups ◽

Order Theory ◽

First Order ◽

First Order Theory

We characterize all ordered Abelian groups whose ﬁrst-order theory in the language {+, <, 0} is strongly dependent. The main result of this note was obtained independently by Halevi and Hasson [7] and Farré [5].

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LOGICAL ASPECTS OF CAYLEY-GRAPHS: THE MONOID CASE

International Journal of Algebra and Computation ◽

10.1142/s0218196706003001 ◽

2006 ◽

Vol 16 (02) ◽

pp. 307-340 ◽

Cited By ~ 11

Author(s):

DIETRICH KUSKE ◽

MARKUS LOHREY

Keyword(s):

Cayley Graph ◽

Cayley Graphs ◽

Order Theory ◽

Second Order ◽

Point Of View ◽

Direct Products ◽

Graph Products ◽

Arbitrary Graph ◽

First Order ◽

First Order Theory

Cayley-graphs of monoids are investigated under a logical point of view. It is shown that the class of monoids, for which the Cayley-graph has a decidable monadic second-order theory, is closed under free products. This result is derived from a result of Walukiewicz, stating that the decidability of monadic second-order theories is preserved under tree-like unfoldings. Concerning first-order logic, it is shown that the class of monoids, for which the Cayley-graph has a decidable first-order theory, is closed under arbitrary graph products, which generalize both, free and direct products. For the proof of this result, tree-like unfoldings are generalized to so-called factorized unfoldings. It is shown that the decidability of the first-order theory of a structure is preserved by factorized unfoldings. Several additional results concerning factorized unfoldings are shown.

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Hyper-regular lattice-ordered groups

Journal of Symbolic Logic ◽

10.2307/2275147 ◽

1993 ◽

Vol 58 (4) ◽

pp. 1342-1358 ◽

Cited By ~ 5

Author(s):

Daniel Gluschankof ◽

François Lucas

Keyword(s):

Abelian Groups ◽

General Context ◽

Lattice Ordered Group ◽

Regular Lattice ◽

Ordered Group ◽

First Order ◽

Elementary Class ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Image Position

It is a well-known fact that the notion of an archimedean order cannot be formalized in the first-order calculus. In [12] and [18], A. Robinson and E. Zakon characterized the elementary class generated by all the archimedean, totally-ordered abelian groups (o-groups) in the language 〈+,<〉, calling it the class of regularly ordered or generalized archimedean abelian groups. Since difference (−) and 0 are definable in that language, it is immediate that in the expanded language 〈 +, −, 0, < 〉 the definable expansion of the class of regular groups is also the elementary class generated by the archimedean ones. In the more general context of lattice-ordered groups (l-groups), the notion of being archimedean splits into two different notions: a strong one (being hyperarchimedean) and a weak one (being archimedean). Using the representation theorem of K. Keimel for hyperarchimedean l-groups, we extend in this paper the Robinson and Zakon characterization to the elementary class generated by the prime-projectable, hyperarchimedean l-groups. This characterization is also extended here to the elementary class generated by the prime-projectable and projectable archimedean l-groups (including all complete l-groups). Finally, transferring a result of A. Touraille on the model theory of Boolean algebras with distinguished ideals, we give the classification up to elementary equivalence of the characterized class.We recall that a lattice-ordered group, l-group for short, is a structure

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Boolean sentence algebras: Isomorphism constructions

Journal of Symbolic Logic ◽

10.2307/2273550 ◽

1983 ◽

Vol 48 (2) ◽

pp. 329-338 ◽

Cited By ~ 4

Author(s):

William P. Hanf ◽

Dale Myers

Keyword(s):

Unary Operation ◽

Order Theory ◽

Bernstein Theorem ◽

Operation Symbol ◽

Relation Symbol ◽

Boolean Space ◽

First Order ◽

First Order Theory ◽

Continuous Maps ◽

Model Spaces

AbstractAssociated with each first-order theory is a Boolean algebra of sentences and a Boolean space of models. Homomorphisms between the sentence algebras correspond to continuous maps between the model spaces. To what do recursive homomorphisms correspond? We introduce axiomatizable maps as the appropriate dual. For these maps we prove a Cantor-Bernstein theorem. Duality and the Cantor-Bernstein theorem are used to show that the Boolean sentence algebras of any two undecidable languages or of any two functional languages are recursively isomorphic where a language is undecidable iff it has at least one operation or relation symbol of two or more places or at least two unary operation symbols, and a language is functional iff it has exactly one unary operation symbol and all other symbols are for unary relations, constants, or propositions.

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