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$\newcommand{\A}{\mathfrak{A}}\newcommand{\B}{\mathfrak{B}}\newcommand{\C}{\mathfrak{C}}$ Given $\A$ an infinite totally ordered set such that it is defined as a disjoint union: $\A := \B \sqcup \C$ of two sets. These sets $\B,\C$ are both infinite and totally ordered such that each element of $\B$ are strictly smaller than each element of $\C$. Then I impose these three sets are isomorphics: $\A:=\B\sqcup \C \simeq \B \simeq \C$.

My question is: Does that type of sets have any name? That it is a specific class of ordinal or whatever?

(Sorry for my bad english) EliX

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The order is an idempotent with respect to the sum of linear orders. – Ramiro de la Vega Feb 27 '13 at 16:51

I would call these orders self-similar, since the whole order consists of two copies of itself, in the way of many fractals.

Examples would include the rational line $\langle\mathbb{Q},\lt\rangle$ as well as the Cantor set and many other fractal-like orders.

You asked,

is it a specific class of ordinal or whatever?

Note that no (nonzero) ordinal has your property, since no ordinal is isomorphic to a proper initial segment of itself. Indeed, one may easily iterate the isomorphism of $A$ with $B$ to produce an infinite descending sequence, so if nonempty, it cannot be well-ordered.

It is easy to construct such orders, and all such orders arise by a suitable iteration of the following procedure:

We have the current collection of points in $B$ and $C$, with every point in $B$ below every point in $C$, and we have order-preserving maps from $B\sqcup C$ to $B$ and also to $C$.

  • We add a new point $p$ either to $B$ or $C$.
  • This new point realizes a certain cut in the order we have so far.
  • We extend the maps by considering the image and pre-image of that cut, creating further new points if necessary to realize those cuts, closing under this process.

For example, one can start with one point in $B$, but this causes one to add a point to $C$ to which to map it, which causes one to create the image of that point in $B$, and so on, back and forth. One can control this process by ensuring during the construction that certain cuts will or will not have a least upper bound.

Finally, note that the property can have no consequences as to the purely local nature of the order, since if we have a self-similar order, we can replace each point by a copy of any fixed linear order, and this resulting order will also be self-similar. For example, $\mathbb{Q}$ copies of $L$, for any linear order $L$, is self-similar. But this order is, locally, just like $L$.

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