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What is the meaning of a set being 'stable'? Is this the same as a set being closed under an operation?

To provide some context, the reference I saw is to a set $M$ being 'stable' where there is a law of composition $\circ:M \times M \rightarrow M$. I haven't seen this useage before, is it very common?

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To get a good answer, you probably need to provide some context and/or an example. –  mrf Dec 26 '12 at 23:58
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3 Answers

up vote 3 down vote accepted

What is the meaning of a set being 'stable'? Is this the same as a set being closed under an operation?

To provide some context, the reference I saw is to a set $M$ being 'stable' where there is a law of composition $\circ:M \times M \rightarrow M$. I haven't seen this usage before, is it very common?

Second question:
No I don't think the usage is common. Michael Artin, for example defines a law of composition on a set $S$ to be "any rule for combining pairs $a, b$ of elements of $S$ to get another element, say $p$ of $S$." (E.g., $p = a\circ b$, $p = a\times b$, $p = a+b$).

More formally, he defines a law of composition as a function of two variables, or a map $\circ : S\times S \to S$, where $S\times S$ denotes the product set, whose elements are pairs of $a, b$ of elements of $S$. I do not see that he defines $S$ as being stable, but given this definition, every pair of elements in $S$ (i.e., a pair $(s_1, s_2) \in S\times S$) maps to (and only to) $S$, and hence for $s_1, s_2 \in S, s_3 = s_1\circ s_2 \in S$.

So to answer your first question:
In this sense, yes, I think one can say that a set $S$ is stable under the binary operation (rule of composition) $*$ if for any $a, b \in S$, $a*b \in S$. I.e., stable, in this sense, meaning $S$ is closed under $*$.


Just curious: What text are you using, or referencing?


Disclaimer: I have answered this based on the tag "abstract algebra", and the (not altogether clear) context in which the term "stable" is used. There are other contexts where one uses "stable groups" in finite group theory, but I suspect that is at a more specialized point than where you encountered "law of composition" (aka, a well-defined binary operation on a set closed under that operation), which is usually early on in abstract algebra, when first defining, say, binary structures like monoids, or groups, etc. There are different contexts altogether (outside of abstract algebra), where the term "stable set" comes up.

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Thanks for your answer, I think it is what I was looking for. I found the reference in these rather obscure online lecture notes, but having searched further, it seems that maybe this phrase is more widely used in the term stable subsets, as on this PlanetMath page. –  user50229 Dec 27 '12 at 1:40
    
Yes, I think you got it! –  amWhy Dec 27 '12 at 1:42
    
+1 like an educational Springer text. –  B. S. Feb 9 '13 at 5:30
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There are three well-known meanings of a stable set. They are:

  1. Graph theory: A set of vertices in a graph, no two of which are adjacent.
  2. Dynamical Systems: The set of points leading up to an attractor.
  3. Game Theory: A set of alternatives satisfying the internal and external stability conditions.
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Despite that might be arguable whether it is an algebraic term or not, bear in mind that another use of the term "stable" is in model theory. This refers to stability of a theory.

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