Editing APS

An APS, or '''Addition-to-Product Sequence''' <math>(G,\\Phi)</math> over a monoidal concrete category, is the following:

* For each natural number <math>n</math>, an associated object of the category, 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.
==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 subobject <math>H_n</math> of <math>G_n</math>, such that the image of <math>H_m</math> &times; <math>H_n</math> under <math>\\Phi_{m,n}</math> lies inside <math>H_{m+n}</math>.

===Quotient APS notion===

{{further|[[quotient APS]]}}

A quotient APS is the image of an APS in a homomorphism that is surjective at each member.

==Properties==

===Injectivity===

{{further|[[IAPS]]}}
An APS is termed injective, or is termed an [[IAPS]], if all the block concatenation maps are injective. Typically, we assume another condition for IAPSes, known as [[refinability]].

===Commutativity===

Very few APSes are commutative. Note that a commutative APS cannot be injective.

===Padding-injectivity===

An APS is termed padding-injective, or is termed a [[PIAPS]], if for any fixed <math>g</math> in <math>G_m</math>, the map sending <math>h</math> in <math>G_n</math> to <math>\\Phi_{m,n}(g,h)</math> is injective.

==See also==

* [[AFS]]
* [[APS of groups]]
* [[IAPS]]
* [[sub-APS]]

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