Re: order on the sets

On Sun, 21 Sep 2008, riderofgiraffes wrote:
Consider an order on the interval such that
[a,b] is higher than [c,d] iff b=>c.

If you ask that [a,b] <= [c,d] iff b<=c, then you
get a particular class of partially ordered set called
a semi-order.

Is [a,b] <= [r,s] when b <= r is an (partial) order for { [a,b] | a <= b }?

Reflexive. If [x.y] <= [x,y], then y <= x <= y. No it's not reflexive.

Asymmetry. If [a,b] <= [r,s] <= [a,b], then
b <= r <= s <= a <= b and [a,b] = [a,a] = [r,s].

Transitivity. If [a,b] <= [x,y] <= [r,s], then
b <= x <= y <= r and [a,b] <= [r,s].

No, it's not an order, nor a semi-order for it lacks reflexitivity.

However [a,b] < [r,s] when b < r is an irreflexive order.
If [x,y] < [x,y], then y < x <= y.
Hence [a,b] <= [r,s] when b < r or (a = r, b = s) is an order.

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