Saturated sub-APS
From Apstheory
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| + | {{basicapsdef}} | ||
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| + | {{set-theoretic sub-APS property}} | ||
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| + | ==Definition== | ||
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A [[sub-APS]] <math>H</math> of an [[APS]] <math>(G,\\Phi)</math> is termed a '''saturated sub-APS''' if for any <math>(m,n)</math>, the inverse image via <math>\\Phi_{m,n}</math> of <math>H_{m+n}</math> is precisely <math>H_m</math> × <math>H_n</math>. | A [[sub-APS]] <math>H</math> of an [[APS]] <math>(G,\\Phi)</math> is termed a '''saturated sub-APS''' if for any <math>(m,n)</math>, the inverse image via <math>\\Phi_{m,n}</math> of <math>H_{m+n}</math> is precisely <math>H_m</math> × <math>H_n</math>. | ||
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| + | ==For groups== | ||
For an APS <math>G</math> of groups with a sub-APS <math>H</math>, the following are equivalent: | For an APS <math>G</math> of groups with a sub-APS <math>H</math>, the following are equivalent: | ||
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* The congruence induced by <math>H</math> is a saturated [[APS congruence]]. | * The congruence induced by <math>H</math> is a saturated [[APS congruence]]. | ||
* The [[quotient APS]] is an [[IAPS of groups]]. | * The [[quotient APS]] is an [[IAPS of groups]]. | ||
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Current revision as of 23:22, 25 January 2007
This article gives a basic definition in the APS theory. It is strictly local to the wiki
This article describes a property that can be evaluated for a sub-APS of an APS and uses only set-theoretical properties
[edit] Definition
A sub-APS Failed to parse (Can't write to or create math temp directory): H
of an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi) is termed a saturated sub-APS if for any Failed to parse (Can't write to or create math temp directory): (m,n)
, the inverse image via Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}
of Failed to parse (Can't write to or create math temp directory): H_{m+n}
is precisely Failed to parse (Can't write to or create math temp directory): H_m
× Failed to parse (Can't write to or create math temp directory): H_n
.
[edit] For groups
For an APS Failed to parse (Can't write to or create math temp directory): G
of groups with a sub-APS Failed to parse (Can't write to or create math temp directory): H
, the following are equivalent:
- Failed to parse (Can't write to or create math temp directory): H
is a saturated sub-APS of Failed to parse (Can't write to or create math temp directory): G
.
- The left congruence induced by Failed to parse (Can't write to or create math temp directory): H
is a saturated APS relation.
- The coset space APS of Failed to parse (Can't write to or create math temp directory): H
in Failed to parse (Can't write to or create math temp directory): G is an IAPS (of sets)
Further, the following are equivalent:
- Failed to parse (Can't write to or create math temp directory): H
is a saturated normal sub-APS of Failed to parse (Can't write to or create math temp directory): G
.
- The congruence induced by Failed to parse (Can't write to or create math temp directory): H
is a saturated APS congruence.
- The quotient APS is an IAPS of groups.
