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fix (transformation action) (Definition)

Let $ A$ be a set, and $ T:A\rightarrow A$ a transformation of that set. We say that $ x\in A$ is fixed by $ T$, or that $ T$ fixes $ x$, whenever

$\displaystyle T(x)=x.$
The subset of fixed elements is called the fixed set of $ T$, and is frequently denoted as $ A^T$.

We say that a subset $ B\subset A$ is fixed by $ T$ whenever all elements of $ B$ are fixed by $ T$, i.e.

$\displaystyle B\subset A^T.$
If this is so, $ T$ restricts to the identity transformation on $ B$.

The definition generalizes readily to a family of transformations with common domain

$\displaystyle T_i : A\rightarrow A,\quad i\in I$
In this case we say that a subset $ B\subset A$ is fixed, if it is fixed by all the elements of the family, i.e. whenever
$\displaystyle B\subset \bigcap_{i\in I} A^{T_i}.$



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"fix (transformation action)" is owned by rmilson. [ full author list (2) ]
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See Also: invariant, transformation, fix

Other names:  fix, fixed, fixes
Also defines:  fixed set
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Cross-references: domain, identity, subset, transformation
There are 41 references to this entry.

This is version 12 of fix (transformation action), born on 2002-02-22, modified 2007-04-15.
Object id is 2510, canonical name is Fixed.
Accessed 13116 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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