Totally ordered partition of a set

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In a totally ordered set, creating a noncrossing partition suggests that each block of the partition inherits the total order property. It is necessary to prove that each block is totally ordered, which can be done using the definition of total order. The discussion emphasizes that if X is totally ordered and A is a subset of X, then A remains totally ordered. The proof process involves demonstrating that for any two elements in A, one must be less than or equal to the other. Overall, the relationship between total orders and noncrossing partitions is affirmed through logical reasoning.
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If I have a totally ordered set and then create a noncrossing partition of that set it seems intuitively obvious that each block of the partition would be totally ordered as well. Can I assume this inheritance or do I need to prove each block is totally ordered? How would one go about proving that if it is the case.
 
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In general, if X is totally ordered and if A\subseteq X, then A is totally ordered.
The proof is not difficult, just use the definition of total order.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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