R-minimal element

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 $x R a$ . 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 $x R a$ 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.

Title	R-minimal element
Canonical name	RminimalElement
Date of creation	2013-03-22 12:42:43
Last modified on	2013-03-22 12:42:43
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	11
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03B10
Synonym	R-minimal
Synonym	$R$ -minimal
Related topic	WellFoundedRelation