Journal article
Sublattices of regular elements
Periodica Mathematica Hungarica, Vol.44(1), pp.111-126
03/2002
DOI: 10.1023/A:1014932204184
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.
Details
- Title: Subtitle
- Sublattices of regular elements
- Creators
- D Anderson - Department of Mathematics The University of Iowa Iowa City IA 52242 U.S.AE Johnson - Department of Mathematics The University of Iowa Iowa City IA 52242 U.S.ARichard Spellerberg II - Department of Mathematics Simpson College Indianola IA 50125 U.S.A
- Resource Type
- Journal article
- Publication Details
- Periodica Mathematica Hungarica, Vol.44(1), pp.111-126
- Publisher
- Kluwer Academic Publishers; Dordrecht
- DOI
- 10.1023/A:1014932204184
- ISSN
- 0031-5303
- eISSN
- 1588-2829
- Language
- English
- Date published
- 03/2002
- Academic Unit
- Mathematics
- Record Identifier
- 9983985830702771
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