IAPS of groups

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This article gives a basic definition in the following area: APS theory
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This article defines the notion of group object in the category of IAPSs|View other types of group objects

Definition

An IAPS of groups is an IAPS over the category of groups. More specifically an IAPS $(G,\Phi )$ is the following data:

For each natural number $n$ , a group denoted $G_{n}$
For each ordered pair $(m,n)$ of natural numbers, an injective homomorphism $\Phi _{m,n}:G_{m}\times G_{n}\to G_{m+n}$

Satisfying the following compatibility conditions:

For $g,h,k$ in $G_{m},G_{n},G_{p}$ respectively:

$\Phi _{m+n,p}(\Phi _{m,n}(g,h),k)=\Phi _{m,n+p}(g,\Phi _{n,p}(h,k))$ .

The above condition is termed an associativity condition.

We may assume $G_{0}$ as the trivial group and define $\Phi _{m,0}$ and $\Phi _{0,n}$ as trivial paddings.

Note that if we remove the condition of injectivity, we get an APS of groups.

Constructions

Sub-IAPS

Further information: sub-IAPS of groups

Let $(G,\Phi )$ be an IAPS of groups. A sub-IAPS $H$ associated to every $n$ a subgroup $H_{n}$ of $G_{n}$ such that the image of $H_{m}\times H_{n}$ under $\Phi _{m,n}$ lies inside $H_{m+n}$ . Note that this is the same as a sub-APS of groups because the injectivity condition comes for free.