Saturated sub-APS
From Apstheory
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| - | A sub-APS <math>H</math> of an APS <math>(G,\\Phi)</math> is termed a | + | 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 an APS <math>G</math> of groups with a sub-APS <math>H</math>, the following are equivalent: | ||
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| + | * <math>H</math> is a saturated sub-APS of <math>G</math>. | ||
| + | * The [[left congruence]] induced by <math>H</math> is a [[saturated APS relation]]. | ||
| + | * The [[coset space APS]] of <math>H</math> in <math>G</math> is an [[IAPS]] (of sets) | ||
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| + | Further, the following are equivalent: | ||
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| + | * <math>H</math> is a saturated [[normal sub-APS]] of <math>G</math>. | ||
| + | * The congruence induced by <math>H</math> is a saturated [[APS congruence]]. | ||
| + | * The [[quotient APS]] is an [[IAPS of groups]]. | ||
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| + | [[Category: Sub-APS properties]] | ||
| + | [[Category: Terminology local to the wiki]] | ||
Revision as of 10:34, 25 December 2006
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
.
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.
