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R-minimal element (Definition)

Let $ A$ be a set and $ R$ be a relation on $ A$. Suppose that $ B$ is a subset of $ A$. An element $ a\in B$ is said to be $ R$-minimal in $ B$ if and only if there is no $ x\in B$ such that $ xRa$. An $ R$-minimal element in $ A$ is simply called $ R$-minimal.

From this definition, it is evident that if $ A$ has an $ R$-minimal element, then $ R$ is not reflexive. However, the definition of $ R$-minimality is sometimes adjusted slightly so as to allow reflexivity: $ a\in B$ is $ R$-minimal (in $ B$) iff the only $ x\in B$ such that $ xRa$ is when $ x=a$.

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.



"R-minimal element" is owned by CWoo. [ full author list (2) | owner history (1) ]
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See Also: well-founded relation

Other names:  R-minimal, $R$-minimal
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Cross-references: minimal, partial order, easy to see, iff, reflexivity, Reflexive, subset, relation
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This is version 8 of R-minimal element, born on 2002-06-02, modified 2008-04-02.
Object id is 2998, canonical name is RMinimalElement.
Accessed 2091 times total.

Classification:
AMS MSC: 03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic)
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