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Weakly least integer closed groups

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Abstract

An archimedean lattice-ordered groupA with distinguished weak unit has the canonical Yosida representation as an ℓ-group of extended real-valued functions on a certain compact Hausdorff spaceY A. Such an ℓ-groupA is calledleast integer closed, orLIC (resp.,weakly least integer closed, orwLIC) if, in the representation,a ∈A implies [a] ∈A (resp., there isa′ ∈A witha′=[a] on a dense set inY A), where [r] ≡ the least integer greater than or equal tor. Earlier, we have studiedLIC groups, with an emphasis on their a-extensions. Here, we turn towLIC groups: we give an intrinsic (though awk-ward) characterization in terms of existence of certain countable suprema. This results also in an intrinsic characterization ofLIC, previously lacking. Also,wLIC is a hull class (whichLIC is not), and the hullwlA is “somewhere near” the projectable hullpA. The best comparison comes from a (somewhat novel) factoringpA=loc(wpA), wherewpA is the “weakly projectable” hull (defined here), andlocB is the “local monoreflection”; then,wpA≤wlA≤loc(wpA), andpA≤loc(wlA), while with a strong unit, all these coincide. Numerous examples and special cases are examined.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Wesleyan University, 06479-0128, Middletown, CT, USA

    Anthony W. Hager

  2. Department of Mathematics, Lafayette College, 18042, Easton, PA, USA

    Chawne M. Kimber

  3. Department of Mathematics and Statistics, Bowling Green State University, 43403, Bowling Green, OH, USA

    Warren Wm. McGovern

Authors
  1. Anthony W. Hager
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  2. Chawne M. Kimber
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  3. Warren Wm. McGovern
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Hager, A.W., Kimber, C.M. & McGovern, W.W. Weakly least integer closed groups. Rend. Circ. Mat. Palermo 52, 453–480 (2003). https://doi.org/10.1007/BF02872765

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