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For a set X of archimedean o-groups, the class of normal-valued ℓ-groups having the property that every value quotient is in X is a complete torsion class. A question of Conrad whether the class is a product torsion class is partially answered by investigating values and value quotients of products of normal-valued ℓ-groups. A major result is that the conjecture that all outer value quotients must be Dedekind complete is equivalent to the conjecture that measurable cardinals do not exist. Received January 20, 2007; accepted in final form February 22, 2008.  相似文献   

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In this paper the relationship between essential extensions and radical classes of lattice-ordered groups is discussed. It is proved that the polar radical classes of lattice-ordered groups are exactly essentially closed classes.Received May 7, 1999; accepted in final form September 23, 2004.  相似文献   

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Aradical class of lattice-ordered groups (-groups) is a class closed under taking convex-subgroups, joins of convex-subgroups, and-isomorphic images. Imposing various other closure conditions leads to many specific types of radical classes (e.g., torsion classes). For several of these types, the complete latticeT of radical classes of that type has been studied, and such latticesT are our object of study here. We give the characteristic properties of closed-kernel radical mappings and polar kernel radical mappings. We prove in many instances thatT isrelatively polarized, that is, for any ], T with ] there exists a unique largest T such that = ], and often we are able to explicitly identify. By using these properties we characterize meet irreducibility in the latticeT of polar kernel radical classes.Presented by M. Henriksen.  相似文献   

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The containmentS nL n is proper. (Here n2 is a positive integer,S n is the small variety of lattice-ordered groups of Scrimger andL n is the variety of lattice-ordered groups defined by the law anbn=bnan.) Jo E. Smith proved this result for n a composite integer. In this note we proveS nL n when n is prime.Presented by L. Fuchs.  相似文献   

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On the atomic conditions of lattice-ordered groups   总被引:2,自引:0,他引:2  
We introduce large convex -subgroups to study the structure of the lattice-ordered groups G whose C(G), P(G) and (G) satisfy atomic conditions, where C(G), P(G) and (G) denote respectively the lattice of all convex -subgroups, the lattice of all polar subgroups and the root system of all regular subgroups of G. In particular, we construct a new torsion class defined as the class of -groups G for which all large prime subgroups are maximal. We prove that the class of hyperarchimedean -groups is properly contained within and that any -group within has the property that any chain of prime subgroups has length at most 2.Received October 7, 2003; accepted in final form June 11, 2004.  相似文献   

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Translated from Matematicheskie Zametki, Vol. 45, No. 1, pp. 72–79, January, 1989.  相似文献   

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Using the set theoretical principle ? for arbitrary large cardinals κ, arbitrary large strongly κ-free abelian groupsA are constructed such that Hom(A, G)={0} for all cotorsion-free groupsG with |G|<κ. This result will be applied to the theory of arbitrary torsion classes for Mod-Z. It allows one, in particular, to prove that the classF of cotorsion-free abelian groups is not cogenerated by aset of abelian groups. This answers a conjecture of Göbel and Wald positively. Furthermore, arbitrary many torsion classes for Mod-Z can be constructed which are not generated or not cogenerated by single abelian groups.  相似文献   

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An Engel l-group generating a proper normal-valued l-variety is shown to be o-approximable. It is also established that for every proper normal-valued l-varietyF, the class (F) of Engel l-groups from F is a torsion class.Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 398–404, July-August, 1995.  相似文献   

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