Periodica Mathematica Hungarica
, Volume 44, Issue 1, pp 111126
First online:
Sublattices of regular elements
 D. D. AndersonAffiliated withDepartment of Mathematics, The University of Iowa
 , E. W. JohnsonAffiliated withDepartment of Mathematics, The University of Iowa
 , Richard L. Spellerberg IIAffiliated withDepartment of Mathematics, Simpson College
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Let L be an rlattice, 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 \(X \leqslant A{\text{ with }}\left. {(0:X) = 0} \right\},{\text{ and }}L_{rg} = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \rightA = \bigvee _\alpha X_\alpha {\text{ where each }}X_\alpha \) 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.
 Title
 Sublattices of regular elements
 Journal

Periodica Mathematica Hungarica
Volume 44, Issue 1 , pp 111126
 Cover Date
 200203
 DOI
 10.1023/A:1014932204184
 Print ISSN
 00315303
 Online ISSN
 15882829
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Authors

 D. D. Anderson ^{(1)}
 E. W. Johnson ^{(1)}
 Richard L. Spellerberg II ^{(2)}
 Author Affiliations

 1. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242, U.S.A.
 2. Department of Mathematics, Simpson College, Indianola, IA, 50125, U.S.A.