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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 $h_n\cdot h\in H_n$. 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 subgroup of $GL(n,R)$.

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

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  • $\begingroup$ To clarify: In Example 1, you want $H_n = 1 + (x^n)$, right? $\endgroup$
    – Neil Epstein
    Commented Feb 25, 2018 at 15:17
  • $\begingroup$ @neil-epstein : right $\endgroup$
    – Dmitry Kerner
    Commented Feb 25, 2018 at 19:36
  • $\begingroup$ I don't know. But I guess you could restate the condition to say that you have a filtered monoid such that $H_j / H_{j+1}$ is a group for all $j$. Interesting condition. Definitely a new one to me. $\endgroup$
    – Neil Epstein
    Commented Feb 27, 2018 at 17:50
  • $\begingroup$ Such structures appear naturally in Singularity Theory, Local Geometry, Comm.Alg. over local rings. (In some cases Nakayama turns them into ordinary groups.) I would expect these to be well known, axiomatized, text-booked. :( $\endgroup$
    – Dmitry Kerner
    Commented Feb 28, 2018 at 13:35

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