Saturated sub-APS

From Apstheory

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.