APS
From Apstheory
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 X 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 |
Terminology
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.
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.
Properties
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.
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 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 <math\\Phi_{m,n}(g,h)</math> is injective.
