Applied Categorical Structures

, Volume 3, Issue 3, pp 261–278 | Cite as

Regular and normal closure operators and categorical compactness for groups

  • Temple H. Fay
  • Gary L. Walls
Article
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Abstract

For a class of groupsF, closed under formation of subgroups and products, we call a subgroupA of a groupG F-regular provided there are two homomorphismsf, g: G » F, withF εF, so thatA = {x εG |f(x) =g(x)}.A is calledF-normal providedA is normal inG andG/A εF. For an arbitrary subgroupA ofG, theF-regular (respectively,F-normal) closure ofA inG is the intersection of allF-regular (respectively,F-normal) subgroups ofG containingA. This process gives rise to two well behaved idempotent closure operators.

A groupG is calledF-regular (respectively,F-normal) compact provided for every groupH, andF-regular (respectively,F-normal) subgroupA ofG × H, π2(A) is anF-regular (respectively,F-normal) subgroup ofH. This generalizes the well known Kuratowski-Mrówka theorem for topological compactness.

In this paper, theF-regular compact andF-normal compact groups are characterized for the classesF consisting of: all torsion-free groups, allR-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.

Mathematics subject classifications (1991)

20J40 20E06 18A99 

Key words

closure operator categorically compact isolator torsion-free 
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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Temple H. Fay
    • 1
  • Gary L. Walls
    • 1
  1. 1.Department of MathematicsUniversity of Southern MississippiHattiesburgUSA

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