irreflexive

A binary relation $\mathcal{R}$ on a set $A$ is said to be irreflexive (or antireflexive) if $\forall a\in A$ , $\neg a\mathcal{R}a$ . In other words, “no element is $\mathcal{R}$ -related to itself.”

For example, the relation $<$ (“less than”) is an irreflexive relation on the set of natural numbers.

Note that “irreflexive” is not simply the negation of “reflexive (http://planetmath.org/Reflexive) .” Although it is impossible for a relation (on a nonempty set) to be both reflexive (http://planetmath.org/Reflexive) and irreflexive, there exist relations that are neither. For example, the relation $\{(a,a)\}$ on the two element set $\{a,b\}$ is neither reflexive nor irreflexive.

Here is an example of a non-reflexive, non-irreflexive relation “in nature.” A subgroup in a group is said to be self-normalizing if it is equal to its own normalizer. For a group $G$ , define a relation $\mathcal{R}$ on the set of all subgroups of $G$ by declaring $H\mathcal{R}K$ if and only if $H$ is the normalizer of $K$ . Notice that every nontrivial group has a subgroup that is not self-normalizing; namely, the trivial subgroup $\{e\}$ consisting of only the identity. Thus, in any nontrivial group $G$ , there is a subgroup $H$ of $G$ such that $\neg H\mathcal{R}H$ . So the relation $\mathcal{R}$ is non-reflexive. Moreover, since the normalizer of a group $G$ in $G$ is $G$ itself, we have $G\mathcal{R}G$ . So $\mathcal{R}$ is non-irreflexive.

Title	irreflexive
Canonical name	Irreflexive
Date of creation	2013-03-22 15:41:45
Last modified on	2013-03-22 15:41:45
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	14
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03E20
Synonym	antireflexive
Related topic	Reflexive