Order: Narrow

[Xittenn]

Where [math] A [/math] is a class, [math] R [/math] a relation and x, y, z are all sets

 

 

"[math] R [/math] is left-narrow" stands for [math] \forall x \left ( \left \{ y \; | \; yRx \right \} \; is \; a \; set \right ) [/math]; "[math] R [/math] is left-narrow on [math] A [/math]" stands for [math] \left ( \forall x \in A \right )\left ( \left \{ y \in A \; | \; yRx \right \} is \; a \; set \right ) [/math]; "[math] R [/math] is right-narrow" stands for [math] \forall x \left ( \left \{ y \; | \; xRy \right \} is \; a \; set \right) [/math]; and "[math] R [/math] is right-narrow on [math] A [/math]" stands for [math] \left ( \forall x \in A \right )\left ( \left \{ y \in A \; | \; xRy \right \} is \; a \; set \right ) [/math]."

 

"[math] R [/math] well orders [math] A [/math]" stands for "[math] R [/math] orders [math] A [/math], and every non-void subset z of [math] A [/math] has an [math] R [/math]-minimal member, and [math] R [/math] is left-narrow on [math] A [/math]" ( ie., for every [math] x \in A [/math] the class of all members of [math] A [/math] which "come before x" is a set."

 

 

Is there another title for the concept of left and right-narrow? A search for left and right-narrow returns no relevant results other than the book that I'm reading and I wish to read other interpretations of the concepts that are along the same line; this being the concept of order and how left and right-narrow are relevant. I guess I'm missing the point on this one! I had skimmed over narrow rather quickly and had somehow taken it to be left and right restriction which it isn't.

 

The development of order is not entirely intuitive as it is being presented in this book. This is in contrast to stating "Every nonempty set of positive integers contains a smallest member." and then proceeding with examples such as developing the Division Algorithm. I need other sources to help me put it all into perspective and finding them is problematic.

 

What do I think this says? Well if functions are right narrow, one-one functions are both and membership is left-narrow but not right this says to me that narrow defines order as non-symmetric as is made a requirement of the text in saying that all orderings are left-narrow for simplicity, I think maybe :/

Edited February 20, 2011 by Xittenn