IAPS with unique class factorization

Definition

An IAPS with unique class factorization $(G,\Phi )$ is an IAPS of groups with the following additional structure/conditions:

• It is an inner-permutative APS. In other words, there is a homomorphism to it from the permutation IAPS, such that the induced action makes it a permutative APS.
• The corresponding conjugacy class APS is cancellative, in the sense that if $\Phi _{m,n}(g,h)$ and $\Phi _{m,n}(g,h')$ are conjugate, so are $h$ and $h'$.
• Call the conjugacy class of $g\in G_{m}$ indecomposable if no element in this conjugacy class can be expressed as the image of a block concatenation map. Then, for every $m$ and every $g\in G_{m}$, $g$ is conjugate to the concatenation of indecomposable elements, and this concatenation is uniquely determined up to ordering and conjugacy for these elements.

In other words, every element can, up to conjugacy, be expressed as a block concatenation of indecomposable elements, and the conjugacy classes of these indecomposable elements are uniquely determined by the conjugacy class of the original element.

We can reformulate this by constructing a monoid of conjugacy classes under the block concatenation map, and then stating that this monoid is an Abelian monoid with unique factorization.