Editing APS of rings

An APS of rings is an [[APS]] over the category of rings. More specifically, an APS <math>(G,\\Phi)</math> of rings is the following data:

* For each natural number <math>n</math>, a ring, denoted <math>G_n</math>.
* For each ordered pair <math>(m,n)</math> of natural numbers, a homomorphism <math>\\Phi_{m,n}:G_m X G_n</math> &rarr; <math>G_{m+n}</math>.

Satisfying  the following compatibility conditions:

For <math>g, h, k</math> in <math>G_m, G_n, G_p</math> respectively, 
<math>\\Phi_{m+n,p} (\\Phi_{m,n}(g,h),k) = \\Phi_{m,n+p} (g, \\Phi_{n,p}(h,k))</math>.

The above condition is termed an associativity condition.

We may assume <math>G_0</math> as the trivial group and define <math>\\Phi_{m,0}</math> and <math>\\Phi_{0,n}</math> as trivial paddings.

==Terminology==

===Members and elements===

For an APS <math>(G,\\Phi)</math> the member <math>G_n</math> is termed the <math>n^{th}</math> member of the APS. A '''member''' of the APS is an object that is the <math>n^{th}</math> member for some <math>n</math>.

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 <math>n</math> for which it is the <math>n^{th}</math> member.

===Block concatenation map===

The maps <math>\\Phi_{m,n}</math> are termed '''block concatenation maps'''.

===Ground member===

The ground member of an APS is its first member.

==Other notions==

===Homomorphism of APSes===

{{further|[[APS homomorphism]]}

Given APSes <math>(G,\\Phi)</math> and <math>(H,\\Psi)</math>, a homomorphism <math>h: G</math> &rarr; <math>H</math> associates, to each natural number <math>n</math>, a map <math>h_n: G_n</math> &rarr; <math>H_n</math>, such that:

<math>\\Psi_{m,n}(h_m(a),h_n(b)) = h_{m+n}(\\Phi_{m,n}(a,b))</math>

===Sub-APS notion===

{{further|[[sub-APS]]}}

Given an APS <math>(G,\\Phi)</math>, a sub-APS <math>H</math> associates, to each <math>n</math>, a subgroup <math>H_n</math> of <math>G_n</math>, such that the image of <math>H_m X H_n</math> under <math>\\Phi_{m,n}</math> lies inside <math>H_{m+n}</math>.

When the APS of groups is injective, any sub-APS is also injective.

===Quotient APS notion===

{{further|[[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.

[[Category: Terminology local to the wiki]]
[[Category: Basic APS definitions]]