APS of groups
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This article gives a basic definition in the following area: APS theory
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Definition
An APS of groups is an APS over the category of groups. More specifically, an APS of groups is the following data:
- For each natural number
, a group, denoted
.
- For each ordered pair
of natural numbers, a homomorphism
.
Satisfying the following compatibility conditions:
For in
respectively:
.
The above condition is termed an associativity condition.
We may assume as the trivial group and define
and
as trivial paddings.
Terminology
Members and elements
For an APS the member
is termed the
member of the APS. A member of the APS is an object that is the
member for some
.
An element of the APS is an element of some member of the APS.
The home of an element of the APS is the member in which it lies. The index of a member is the for which it is the
member.
Block concatenation map
The maps are termed block concatenation maps.
Ground member
The ground member of an APS is its first member.
Other notions
Homomorphism of APSes
Further information: APS homomorphism
Given APSes and
, a homomorphism
→
associates, to each natural number
, a map
→
, such that:
Sub-APS notion
Further information: sub-APS
Given an APS , a sub-APS
associates, to each
, a subgroup
of
, such that the image of
under
lies inside
.
When the APS of groups is injective, any sub-APS is also injective.
Quotient APS notion
Further information: quotient APS
A quotient APS is the image of an APS in a homomorphism that is surjective at each member.
Normal sub-APSes, kernels and images
Given a homomorphism of APSes of groups, the kernels of the individual homomorphisms for a sub-APS of the domain APS, and the images of the individual homomorphism form a sub-APS of the range APS. The image is thus a quotient APS.
Further, we have the following result: a sub-APS of an APS of groups occurs as the kernel of an APS homomorphism if and only if every member of it is a normal subgroup of the corresponding member of the whole APS. A sub-APS satisfying either of these equivalent conditions is termed a normal sub-APS.
This parallels the group theory result that a subgroup of a group occurs as the kernel of a group homomorphism if and only if it is normal.
Properties
Injectivity
An APS of groups is termed injective, or an IAPS of groups, if every block concatenation map is injective. For an IAPS of groups, we usually also assume the condition of refinability.
Commutativity
Very few APSes of groups are commutative. Note that a commutative APS cannot also be injective.
Padding-injectivity
Most APSes of groups that we encounter satisfy the condition of being padding-injective.