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Periodica Mathematica Hungarica

, Volume 44, Issue 1, pp 111–126 | Cite as

Sublattices of regular elements

  • D. D. Anderson
  • E. W. Johnson
  • Richard L. Spellerberg II
Article
  • 39 Downloads

Abstract

Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: \(L_f = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|\left. {(0:A) = 0} \right\},L_{sr} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|\) there is a compact element \(X \leqslant A\) with \(\left. {(0:X) = 0} \right\},L_r = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|\) there is a principal element \(X \leqslant A{\text{ with }}\left. {(0:X) = 0} \right\},{\text{ and }}L_{rg} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|A = \bigvee _\alpha X_\alpha {\text{ where each }}X_\alpha \) is a principal element with (0:X_{\alpha })=0\} . The first three subsets L_{f}, L_{sr}, and L_{r} are augmented filters \(\mathcal{L}^0 \) on L, i.e., \(\mathcal{L}^0 = \mathcal{L} \cup \left\{ 0 \right\}{\text{ where }}\mathcal{L}\) is a multiplicatively closed subset of $L$ with $A\in \(\mathcal{L}\) and $B\geq A$ with $B\in L$ implies $B\in \(\mathcal{L}\) and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module.

Keywords

General Situation Regular Element Great Element Principal Element Compact Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • D. D. Anderson
    • 1
  • E. W. Johnson
    • 1
  • Richard L. Spellerberg II
    • 2
  1. 1.Department of MathematicsThe University of IowaIowa CityU.S.A.
  2. 2.Department of MathematicsSimpson CollegeIndianolaU.S.A.

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