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Consider this sequence:

$(...,1,1,1,\overbrace{1,0,0}^\prime,1,0,0,0,0,0,0,0,0,1,0,0,\overbrace{0,0,0,,....,0,0,0, \;}^{24} 1,0,0,0,0,0,0,0,0,...)$

Let us partition this sequence into ordered subsequences of three elements starting with the three elements indicated by the prime. Let us make this substitution for each such subsequence: $(1,0,0) \to 1$ and $(0,0,0) \to 0$. Then this sequence collapses into itself. Note that this collapse does not work unless we have the infinite sequence of 1's on the left

I would like to call the above sequence "self-similar". Is there a way to extend the definition of self-similar to sequences?

In my research, I'm looking at an object which is considerably more sophisticated, but this simple sequence gets the idea across.

As a follow up question, this idea could be extended to a matrix $A_{i,j}$ where $\infty < i,j <\infty$, although writing such a matrix in matrix form would take up a lot of space. What notation might work?

p.s. A similar question was asked before, but that question was closed because it was vague.

Chris
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  • You need to review what a set is. What you wrote above is the same set as ${0,1}$. – JRN Dec 13 '22 at 01:30
  • Oops. I'm correcting it now. – Chris Dec 13 '22 at 01:31
  • It's probably worth rephrasing it as a substitution: $1\to 100,$ $0\to 000.$ – Thomas Andrews Dec 13 '22 at 01:47
  • You are not talking about ordered sets (sets famously do not have multiplicity), you are talking about strings of digits. Your description would have been so much easier to understand if you just said what Thomas said. Why are there infinitely many $1$s in the beginning; doesn't that break the pattern? What is $\infty<i,j<\infty$ supposed to mean? Either "self-similar" has a relevant technical definition, or it doesn't. If it does, provide it, else your question is a subjective yes/no question. – anon Dec 13 '22 at 01:56
  • How about ordered list? I made the change suggested by Thomas. With regard to the infinite 1s at the beginning, I believe that they must be there in order for the collapse I described in my question to work. I guess I'm asking whether there is a relevant definition of self-similar that applies to sets. – Chris Dec 13 '22 at 02:03
  • I guess I can make up my own definition of a self-similar set. However, I was wondering if someone else has done this already? – Chris Dec 13 '22 at 02:11
  • @Chris: The term you want is "sequence"; each group of three is a "subsequence". (BTW: The $1$s you've added to the left are a bit of a distraction. I'd suggest restoring your original sequence to set the stage for your question, and then mention later that the interesting property is preserved when the $1$s are added.) ... I doubt "self-similar" is really the best name for the property, as the term usually has a geometric connotation. Something like "self-collapsing" ("self-origamic", due to folding?) might better convey your rule. – Blue Dec 13 '22 at 02:34
  • I changed everything to sequence. I believe that the infinite 1s on the left are not just an interesting property. I think the must be there for this sequence to be "self-collapsing". Please ponder the collapsing operation and confirm that the 1s on the left must be there. I could be wrong. – Chris Dec 13 '22 at 02:39
  • @Blue I realize that the term self-similar has geometric connotations, but perhaps we generalize it to sequences? – Chris Dec 13 '22 at 15:33

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