Sublattices of regular elements
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”: $$L_f = \left\{ 0 \right\} \cup \left\{ A \right. \in \left. L \right|\lef...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Anderson, D. D. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2002 |
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| Schlagwörter: |
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| Anmerkung: |
© Kluwer Academic Publishers 2002 |
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| Übergeordnetes Werk: |
Enthalten in: Periodica mathematica Hungarica - Kluwer Academic Publishers, 1971, 44(2002), 1 vom: März, Seite 111-126 |
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| Übergeordnetes Werk: |
volume:44 ; year:2002 ; number:1 ; month:03 ; pages:111-126 |
| Links: |
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| DOI / URN: |
10.1023/A:1014932204184 |
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| Katalog-ID: |
OLC2036923062 |
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| 520 | |a Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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}$$ and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module. | ||
| 650 | 4 | |a General Situation | |
| 650 | 4 | |a Regular Element | |
| 650 | 4 | |a Great Element | |
| 650 | 4 | |a Principal Element | |
| 650 | 4 | |a Compact Element | |
| 700 | 1 | |a Johnson, E. W. |4 aut | |
| 700 | 1 | |a Spellerberg II, Richard L. |4 aut | |
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10.1023/A:1014932204184 doi (DE-627)OLC2036923062 (DE-He213)A:1014932204184-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Anderson, D. D. verfasserin aut Sublattices of regular elements 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2002 Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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}$$ and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module. General Situation Regular Element Great Element Principal Element Compact Element Johnson, E. W. aut Spellerberg II, Richard L. aut Enthalten in Periodica mathematica Hungarica Kluwer Academic Publishers, 1971 44(2002), 1 vom: März, Seite 111-126 (DE-627)129291307 (DE-600)120494-4 (DE-576)014472597 0031-5303 nnns volume:44 year:2002 number:1 month:03 pages:111-126 https://doi.org/10.1023/A:1014932204184 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 AR 44 2002 1 03 111-126 |
| spelling |
10.1023/A:1014932204184 doi (DE-627)OLC2036923062 (DE-He213)A:1014932204184-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Anderson, D. D. verfasserin aut Sublattices of regular elements 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2002 Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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}$$ and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module. General Situation Regular Element Great Element Principal Element Compact Element Johnson, E. W. aut Spellerberg II, Richard L. aut Enthalten in Periodica mathematica Hungarica Kluwer Academic Publishers, 1971 44(2002), 1 vom: März, Seite 111-126 (DE-627)129291307 (DE-600)120494-4 (DE-576)014472597 0031-5303 nnns volume:44 year:2002 number:1 month:03 pages:111-126 https://doi.org/10.1023/A:1014932204184 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 AR 44 2002 1 03 111-126 |
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10.1023/A:1014932204184 doi (DE-627)OLC2036923062 (DE-He213)A:1014932204184-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Anderson, D. D. verfasserin aut Sublattices of regular elements 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2002 Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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}$$ and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module. General Situation Regular Element Great Element Principal Element Compact Element Johnson, E. W. aut Spellerberg II, Richard L. aut Enthalten in Periodica mathematica Hungarica Kluwer Academic Publishers, 1971 44(2002), 1 vom: März, Seite 111-126 (DE-627)129291307 (DE-600)120494-4 (DE-576)014472597 0031-5303 nnns volume:44 year:2002 number:1 month:03 pages:111-126 https://doi.org/10.1023/A:1014932204184 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 AR 44 2002 1 03 111-126 |
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10.1023/A:1014932204184 doi (DE-627)OLC2036923062 (DE-He213)A:1014932204184-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Anderson, D. D. verfasserin aut Sublattices of regular elements 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2002 Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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}$$ and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module. General Situation Regular Element Great Element Principal Element Compact Element Johnson, E. W. aut Spellerberg II, Richard L. aut Enthalten in Periodica mathematica Hungarica Kluwer Academic Publishers, 1971 44(2002), 1 vom: März, Seite 111-126 (DE-627)129291307 (DE-600)120494-4 (DE-576)014472597 0031-5303 nnns volume:44 year:2002 number:1 month:03 pages:111-126 https://doi.org/10.1023/A:1014932204184 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 AR 44 2002 1 03 111-126 |
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10.1023/A:1014932204184 doi (DE-627)OLC2036923062 (DE-He213)A:1014932204184-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Anderson, D. D. verfasserin aut Sublattices of regular elements 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Kluwer Academic Publishers 2002 Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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. General Situation Regular Element Great Element Principal Element Compact Element Johnson, E. W. aut Spellerberg II, Richard L. aut Enthalten in Periodica mathematica Hungarica Kluwer Academic Publishers, 1971 44(2002), 1 vom: März, Seite 111-126 (DE-627)129291307 (DE-600)120494-4 (DE-576)014472597 0031-5303 nnns volume:44 year:2002 number:1 month:03 pages:111-126 https://doi.org/10.1023/A:1014932204184 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2005 GBV_ILN_2088 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 AR 44 2002 1 03 111-126 |
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Enthalten in Periodica mathematica Hungarica 44(2002), 1 vom: März, Seite 111-126 volume:44 year:2002 number:1 month:03 pages:111-126 |
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D.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Sublattices of regular elements</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2002</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Kluwer Academic Publishers 2002</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">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 $$ is a principal element with (0:X_{\alpha })=0\} . 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Sublattices of regular elements |
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Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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}$$ and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module. © Kluwer Academic Publishers 2002 |
| abstractGer |
Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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}$$ and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are “regular” with respect to an $L$-module. © Kluwer Academic Publishers 2002 |
| abstract_unstemmed |
Abstract Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of “regular elements”: $$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. © Kluwer Academic Publishers 2002 |
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