Special-valuedl-groups
Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgrou...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Bixler, J. Patrick [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
1986 |
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| Schlagwörter: |
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| Anmerkung: |
© Birkhäuser Verlag 1986 |
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| Übergeordnetes Werk: |
Enthalten in: Algebra universalis - Birkhäuser-Verlag, 1971, 22(1986), 2-3 vom: Juni, Seite 172-191 |
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| Übergeordnetes Werk: |
volume:22 ; year:1986 ; number:2-3 ; month:06 ; pages:172-191 |
| Links: |
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| DOI / URN: |
10.1007/BF01224024 |
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| Katalog-ID: |
OLC2069277372 |
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| 520 | |a Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. | ||
| 650 | 4 | |a Root System | |
| 650 | 4 | |a Simple Proof | |
| 650 | 4 | |a Direct Proof | |
| 650 | 4 | |a Special Element | |
| 650 | 4 | |a Easy Proof | |
| 700 | 1 | |a Darnel, Michael |4 aut | |
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10.1007/BF01224024 doi (DE-627)OLC2069277372 (DE-He213)BF01224024-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Bixler, J. Patrick verfasserin aut Special-valuedl-groups 1986 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Verlag 1986 Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. Root System Simple Proof Direct Proof Special Element Easy Proof Darnel, Michael aut Enthalten in Algebra universalis Birkhäuser-Verlag, 1971 22(1986), 2-3 vom: Juni, Seite 172-191 (DE-627)129291129 (DE-600)120470-1 (DE-576)014472449 0002-5240 nnns volume:22 year:1986 number:2-3 month:06 pages:172-191 https://doi.org/10.1007/BF01224024 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_24 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4323 AR 22 1986 2-3 06 172-191 |
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10.1007/BF01224024 doi (DE-627)OLC2069277372 (DE-He213)BF01224024-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Bixler, J. Patrick verfasserin aut Special-valuedl-groups 1986 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Verlag 1986 Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. Root System Simple Proof Direct Proof Special Element Easy Proof Darnel, Michael aut Enthalten in Algebra universalis Birkhäuser-Verlag, 1971 22(1986), 2-3 vom: Juni, Seite 172-191 (DE-627)129291129 (DE-600)120470-1 (DE-576)014472449 0002-5240 nnns volume:22 year:1986 number:2-3 month:06 pages:172-191 https://doi.org/10.1007/BF01224024 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_24 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4323 AR 22 1986 2-3 06 172-191 |
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10.1007/BF01224024 doi (DE-627)OLC2069277372 (DE-He213)BF01224024-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Bixler, J. Patrick verfasserin aut Special-valuedl-groups 1986 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Verlag 1986 Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. Root System Simple Proof Direct Proof Special Element Easy Proof Darnel, Michael aut Enthalten in Algebra universalis Birkhäuser-Verlag, 1971 22(1986), 2-3 vom: Juni, Seite 172-191 (DE-627)129291129 (DE-600)120470-1 (DE-576)014472449 0002-5240 nnns volume:22 year:1986 number:2-3 month:06 pages:172-191 https://doi.org/10.1007/BF01224024 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_24 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4323 AR 22 1986 2-3 06 172-191 |
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10.1007/BF01224024 doi (DE-627)OLC2069277372 (DE-He213)BF01224024-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Bixler, J. Patrick verfasserin aut Special-valuedl-groups 1986 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Verlag 1986 Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. Root System Simple Proof Direct Proof Special Element Easy Proof Darnel, Michael aut Enthalten in Algebra universalis Birkhäuser-Verlag, 1971 22(1986), 2-3 vom: Juni, Seite 172-191 (DE-627)129291129 (DE-600)120470-1 (DE-576)014472449 0002-5240 nnns volume:22 year:1986 number:2-3 month:06 pages:172-191 https://doi.org/10.1007/BF01224024 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_24 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4323 AR 22 1986 2-3 06 172-191 |
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10.1007/BF01224024 doi (DE-627)OLC2069277372 (DE-He213)BF01224024-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Bixler, J. Patrick verfasserin aut Special-valuedl-groups 1986 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser Verlag 1986 Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. Root System Simple Proof Direct Proof Special Element Easy Proof Darnel, Michael aut Enthalten in Algebra universalis Birkhäuser-Verlag, 1971 22(1986), 2-3 vom: Juni, Seite 172-191 (DE-627)129291129 (DE-600)120470-1 (DE-576)014472449 0002-5240 nnns volume:22 year:1986 number:2-3 month:06 pages:172-191 https://doi.org/10.1007/BF01224024 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_21 GBV_ILN_24 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2009 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4314 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4323 AR 22 1986 2-3 06 172-191 |
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Enthalten in Algebra universalis 22(1986), 2-3 vom: Juni, Seite 172-191 volume:22 year:1986 number:2-3 month:06 pages:172-191 |
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Enthalten in Algebra universalis 22(1986), 2-3 vom: Juni, Seite 172-191 volume:22 year:1986 number:2-3 month:06 pages:172-191 |
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Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. 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Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. © Birkhäuser Verlag 1986 |
| abstractGer |
Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. © Birkhäuser Verlag 1986 |
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Abstract Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an $ a^{*} $-extension of the originall-group. Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be ν-embedded into a special-valuedl-group. © Birkhäuser Verlag 1986 |
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