A Generalization to Ordered Groups of a Kreĭn Theorem

Generalized Commutativity of Lattice-Ordered Groups

Mathematica Slovaca ◽

10.1515/ms-2015-0026 ◽

2015 ◽

Vol 65 (2) ◽

Cited By ~ 1

Author(s):

M. R. Darnel ◽

W. C. Holland ◽

H. Pajoohesh

Keyword(s):

Theorem Proving ◽

Natural Generalization ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Weak Commutativity

AbstractIn this paper we explore generalizations of Neumann’s theorem proving that weak commutativity in ordered groups actually implies the group is abelian. We show that a natural generalization of Neumann’s weak commutativity holds for certain Scrimger ℓ-groups.

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Intrinsic metric preserving maps on partially ordered groups

Algebra Universalis ◽

10.1007/bf01192713 ◽

1996 ◽

Vol 36 (1) ◽

pp. 135-140 ◽

Cited By ~ 5

Author(s):

M. Jasem

Keyword(s):

Intrinsic Metric ◽

Partially Ordered ◽

Ordered Groups

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ON THE STRUCTURE OF GROUPS ENDOWED WITH A COMPATIBLE C-RELATION

Journal of Symbolic Logic ◽

10.1017/jsl.2018.41 ◽

2018 ◽

Vol 83 (3) ◽

pp. 939-966

Author(s):

GABRIEL LEHÉRICY

Keyword(s):

Ordered Groups ◽

Quasi Order ◽

Minimal Groups

AbstractWe use quasi-orders to describe the structure of C-groups. We do this by associating a quasi-order to each compatible C-relation of a group, and then give the structure of such quasi-ordered groups. We also reformulate in terms of quasi-orders some results concerning C-minimal groups given in [5].

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Finitely generic models of TUH, for certain model companionable theories T

Journal of Symbolic Logic ◽

10.2307/2274316 ◽

1985 ◽

Vol 50 (3) ◽

pp. 604-610

Author(s):

Francoise Point

Keyword(s):

Discriminator Variety ◽

Generic Models ◽

Elementary Class ◽

Ordered Groups ◽

Existentially Closed ◽

Closed Element ◽

Starting Point ◽

Joint Embedding ◽

Joint Embedding Property ◽

Generic Elements

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

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