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| Title of object: |
R-minimal element |
| Canonical Name: |
RMinimalElement |
| Type: |
Definition |
| Created on: |
2002-06-02 10:41:14 |
| Modified on: |
2008-04-02 00:23:08 |
| Classification: |
msc:03B10 |
| Synonyms: |
R-minimal element=R-minimal
R-minimal element=$R$-minimal |
Preamble:
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Content:
Let $A$ be a set and $R$ be a relation on $A$. An element $a\in A$ is said to be {\bf $R$-minimal} if and only if there is no $x\in A$ such that $xRa$. From this definition, it is evident that $R$ is not reflexive. However, the definition of $R$-minimality is sometimes adjusted slightly so as to allow reflexivity: $a\in A$ is $R$-minimal iff the only $x\in A$ such that $xRa$ is when $x=a$.
\textbf{Remark}. Using the second definition, it is easy to see that when $R$ is a partial order, then an element $a$ is $R$-minimal iff it is minimal. |
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