connected relation

A (binary) relation \sim on a set AA is connected if any two elements that are related in neither order are equal:

(x,y:A),xyyxx=y. \forall (x, y: A),\; x \nsim y \;\wedge\; y \nsim x \;\Rightarrow\; x = y .

This is a basic property of linear orders; an apartness relation is usually called tight if it is connected.

Using excluded middle, it is equivalent to say that every two elements are related in some order or equal:

(x,y:A),xyyxx=y. \forall (x, y: A),\; x \sim y \;\vee\; y \sim x \;\vee\; x = y .

However, this version is too strong for the intended applications to constructive mathematics. (In particular, <\lt on the located Dedekind real numbers satisfies the first definition but not this one.)

On the other hand, there is a stronger notion that may be used in constructive mathematics, if AA is already equipped with a tight apartness #\#. In that case, we say that \sim is strongly connected if any two distinct elements are related in one order or the other:

(x,y:A),x#yxyyx. \forall (x, y: A),\; x \# y \;\Rightarrow\; x \sim y \;\vee\; y \sim x .

Since #\# is connected itself, every strongly connected relation is connected; the converse holds with excluded middle (through which every set has a unique tight apartness).

Revised on August 24, 2012 20:05:45 by Urs Schreiber (