Further Properties of the Lattice of Torsion Classes of Abelian Cyclically Ordered Groups

2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Judita Lihová ◽  
Ján Jakubík
Keyword(s):  
Torsion Class ◽  
Complete Distributivity ◽  
Fundamental Properties ◽  
Ordered Groups ◽  
Dual Atom

AbstractThe notion of torsion class of abelian cyclically ordered groups has been introduced and fundamental properties of the collection T of all such classes, ordered by the class-theoretical inclusion, have been proved by the second author in 2011. The present paper can be considered as a continuation of the above mentioned one. We describe all atoms of T , show that T does not have any dual atom and prove complete distributivity of T .

Torsion classes of generalized Boolean algebras

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Analogous Result ◽  
Boolean Algebras ◽  
Radical Class ◽  
Torsion Class ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Brouwerian Lattice ◽  
Generalized Boolean Algebras

AbstractTorsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T g and R g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T g and R g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized Boolean algebras.

Torsion classes of abelian cyclically ordered groups

2012 ◽  
Vol 62 (4) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Distributive Lattice ◽  
Partial Order ◽  
Torsion Class ◽  
Proper Class ◽  
Partially Ordered ◽  
Ordered Groups ◽  
Lattice Ordered Groups

AbstractWe introduce the notion of torsion class of abelian cyclically ordered groups; the definition is analogous to that used in the theory of lattice ordered groups. The collection T of all such classes is partially ordered by the class-theoretical inclusion. Though T is a proper class, we can apply the usual terminology for this partial order. We prove that T is a complete, infinitely distributive lattice having infinitely many atoms.

New torsion theory in unital abelian ℓ-groups

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Jorge Martínez
Keyword(s):  
Torsion Theory ◽  
Torsion Class ◽  
Finite Range ◽  
Order Unit ◽  
Concrete Category ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice ◽  
Strong Order

AbstractThis paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects.Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean latticeordered groups are characterized: for each subgroup A of the rationals with the identity 1, either $\mathfrak{T} = \mathfrak{S}\left( A \right)$, the class of all lattice-ordered groups of functions on a set X which have finite range in A, or $$\mathfrak{T} = \mathbb{T}\left( A \right)$$, the class of all subgroups of A with 1.As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup A of the reals with 1, of rank two or greater, then $\mathfrak{T}$ contains all ℓ-groups of the form $A\vec \times G$, for all abelian lattice-ordered groups G. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\vec \times G$, for all abelian lattice-ordered groups G.

Metastable Defects in the Amorphous Silicon-Germanium Alloys

2003 ◽  
Vol 762 ◽  
Author(s):  
J. David Cohen
Keyword(s):  
Amorphous Silicon ◽  
Silicon Germanium ◽  
Dangling Bonds ◽  
Annealing Behavior ◽  
Fundamental Properties ◽  
Silicon Materials ◽  
Salient Features ◽  
The Individual ◽  
Amorphous Silicon Germanium ◽  
Silicon Germanium Alloys

AbstractThis paper first briefly reviews a few of the early studies that established some of the salient features of light-induced degradation in a-Si,Ge:H. In particular, I discuss the fact that both Si and Ge metastable dangling bonds are involved. I then review some of the recent studies carried out by members of my laboratory concerning the details of degradation in the low Ge fraction alloys utilizing the modulated photocurrent method to monitor the individual changes in the Si and Ge deep defects. By relating the metastable creation and annealing behavior of these two types of defects, new insights into the fundamental properties of metastable defects have been obtained for amorphous silicon materials in general. I will conclude with a brief discussion of the microscopic mechanisms that may be responsible.

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