Unique factorization APS
From Apstheory
Current revision as of 08:13, 26 January 2007
[edit] Definition
A unique factorization APS is an APS of sets with the following properties:
- It is commutative, viz Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n} (a, b) = \\Phi_{n,m} (b, a)
.
- It is cancellative, viz Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}(a, b) = \\Phi_{m,n}(a, b')
implies that Failed to parse (Can't write to or create math temp directory): b = b'
.
- Call an element irreducible if it cannot be expressed as a block concatenation of other elements. Then, every element has a unique expression (upto ordering) as a sum of irreducible elements.
The third condition is the important unique factorization condition.
An alternative way of defining unique factorization APS (for structures with a notion is as follows: given a commutative cancellative APS with a natural trivial element, we can associate with it a monoid which as a set is the set of elements of all members of the APS modulo the trivial padding. Then, the APS is a unique factorization APS if the associated monoid is a unique factorization monoid.
[edit] Typical examples
A typical example of an APS which we would like to prove as satisfying unique factorization is a representation APS of a single group over an APS of groups.
We also like to prove unique factorization for the conjugacy class APS, which is a special case of a representation APS with the single group being the group of integers.
