APS
From Apstheory
m (made math formula in separate line) |
(Changed X to ×) |
||
| (3 intermediate revisions not shown) | |||
| Line 2: | Line 2: | ||
* For each natural number <math>n</math>, an associated object of the category, denoted <math>G_n</math>. | * 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 | + | * For each ordered pair <math>(m,n)</math> of natural numbers, a homomorphism <math>\\Phi_{m,n}:G_m</math> × <math>G_n</math> → <math>G_{m+n}</math>. |
Satisfying the following compatibility conditions: | Satisfying the following compatibility conditions: | ||
| Line 33: | Line 33: | ||
===Homomorphism of APSes=== | ===Homomorphism of APSes=== | ||
| - | {{further|[[APS homomorphism]]} | + | {{further|[[APS homomorphism]]}} |
Given APSes <math>(G,\\Phi)</math> and <math>(H,\\Psi)</math>, a homomorphism <math>h: G</math> → <math>H</math> associates, to each natural number <math>n</math>, a map <math>h_n: G_n</math> → <math>H_n</math>, such that: | Given APSes <math>(G,\\Phi)</math> and <math>(H,\\Psi)</math>, a homomorphism <math>h: G</math> → <math>H</math> associates, to each natural number <math>n</math>, a map <math>h_n: G_n</math> → <math>H_n</math>, such that: | ||
| Line 43: | Line 43: | ||
{{further|[[sub-APS]]}} | {{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 | + | 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> × <math>H_n</math> under <math>\\Phi_{m,n}</math> lies inside <math>H_{m+n}</math>. |
===Quotient APS notion=== | ===Quotient APS notion=== | ||
| Line 64: | Line 64: | ||
===Padding-injectivity=== | ===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. | + | 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== | ==See also== | ||
Current revision as of 10:23, 25 December 2006
An APS, or Addition-to-Product Sequence Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
over a monoidal concrete category, is the following:
- For each natural number Failed to parse (Can't write to or create math temp directory): n
, an associated object of the category, denoted Failed to parse (Can't write to or create math temp directory): G_n .
- For each ordered pair Failed to parse (Can't write to or create math temp directory): (m,n)
of natural numbers, a homomorphism Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}:G_m
× Failed to parse (Can't write to or create math temp directory): G_n
→ Failed to parse (Can't write to or create math temp directory): G_{m+n}
.
Satisfying the following compatibility conditions:
For Failed to parse (Can't write to or create math temp directory): g, h, k
in Failed to parse (Can't write to or create math temp directory): G_m, G_n, G_p respectively:
Failed to parse (Can't write to or create math temp directory): \\Phi_{m+n,p} (\\Phi_{m,n}(g,h),k) = \\Phi_{m,n+p} (g, \\Phi_{n,p}(h,k)) .
The above condition is termed an associativity condition.
Contents |
[edit] Terminology
[edit] Members and elements
For an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
the member Failed to parse (Can't write to or create math temp directory): G_n
is termed the Failed to parse (Can't write to or create math temp directory): n^{th}
member of the APS. A member of the APS is an object that is the Failed to parse (Can't write to or create math temp directory): n^{th}
member for some Failed to parse (Can't write to or create math temp directory): n
.
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 Failed to parse (Can't write to or create math temp directory): n
for which it is the Failed to parse (Can't write to or create math temp directory): n^{th}
member.
[edit] Block concatenation map
The maps Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}
are termed block concatenation maps.
[edit] Ground member
The ground member of an APS is its first member.
[edit] Other notions
[edit] Homomorphism of APSes
- Further information: APS homomorphism
Given APSes Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
and Failed to parse (Can't write to or create math temp directory): (H,\\Psi)
, a homomorphism Failed to parse (Can't write to or create math temp directory): h: G
→ Failed to parse (Can't write to or create math temp directory): H associates, to each natural number Failed to parse (Can't write to or create math temp directory): n
, a map Failed to parse (Can't write to or create math temp directory): h_n: G_n
→ Failed to parse (Can't write to or create math temp directory): H_n
, such that:
Failed to parse (Can't write to or create math temp directory): \\Psi_{m,n}(h_m(a),h_n(b)) = h_{m+n}(\\Phi_{m,n}(a,b))
[edit] Sub-APS notion
- Further information: sub-APS
Given an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi) , a sub-APS Failed to parse (Can't write to or create math temp directory): H
associates, to each Failed to parse (Can't write to or create math temp directory): n
, a subobject Failed to parse (Can't write to or create math temp directory): H_n
of Failed to parse (Can't write to or create math temp directory): G_n
, such that the image of 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
under Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}
lies inside Failed to parse (Can't write to or create math temp directory): H_{m+n}
.
[edit] 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.
[edit] Properties
[edit] Injectivity
- Further information: 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.
[edit] Commutativity
Very few APSes are commutative. Note that a commutative APS cannot be injective.
[edit] Padding-injectivity
An APS is termed padding-injective, or is termed a PIAPS, if for any fixed Failed to parse (Can't write to or create math temp directory): g
in Failed to parse (Can't write to or create math temp directory): G_m
, the map sending 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_n
to Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}(g,h)
is injective.
