Order reversing involution
- From: Frederick Williams <frederick.williams2@xxxxxxxxx>
- Date: Fri, 14 Nov 2008 22:31:30 +0000
Suppose a set X is ordered by <=, i.e.:
(1) x <= x,
(2) x <= y, y <= x imply x = y, and
(3) x <= y, y <= z imply x <= z.
And let n:X -> X be an order reversing involution, i.e.:
(4) x <= y implies ny <= nx, and
(5) nnx = x.
Can one prove
(6) ny <= nx implies x <= y ?
And can one prove (4) and (5) from (6)?
If one can't prove those implications, does it help to assume that X is
linearly ordered, i.e. (1), (2), (3) plus:
(7) (x <= y) or (y <= x) ?
Suppose X is finite, can one prove, using some or all of (1) to (7),
that X has cardinality a power of two?
--
He is not here; but far away
The noise of life begins again
And ghastly thro' the drizzling rain
On the bald street breaks the blank day.
.
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