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Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. (i.e. every element is order-by-order invertible)

If a filtration stabilizes at a finite step to the identity element, then $H$ is a group. In many cases the filtration is infinite, so $H$ is not necessarily a group, and yet close to being a group.

Examples:

  1. Let $R=k[x]\supset (x)\supset (x^2)\supset\cdots$. The elements of $1+(x)$ are order-by-order invertible. If $R$ were local, the elements would be invertible.

  2. $H=\{A|\ \det(A)\in \{1\}+(x)\}\subset Mat_{n\times n}(R)$. If $R$ were local this would be a group.

What is the standard name for such "almost groups"? Some references?

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  • $\begingroup$ Are the $H_i$'s supposed to be submonoids? $\endgroup$
    – J.-E. Pin
    Commented Mar 11, 2018 at 10:00

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