-1
$\begingroup$

For my work I need many of the very easy and basic properties of suprema and infima. While they are all pretty easy to prove, I would prefer to refer to a standard text book. However I did not find one and worse, do not know how to search for one.

I am interested in properties for partially ordered sets like

$A\subseteq B \Rightarrow \inf B \le \inf A$

$\inf \bigcup A=\inf \{\inf a|a\in A\}$

And for sets with a "$+$", and an inverse $-$ (i.e. $-x \le -y \leftrightarrow x\ge y$):

$\inf X=-\sup (-X)$

$\inf (X+m) = (\inf X) +m $

And the equivalent for functions into such sets ($f: Y \rightarrow X$):

$\inf_{y\in Y} f(y)=-\sup_{y\in Y} -f(y)$

and so on...

Is there like a "reference handbook"?

$\endgroup$
1
  • $\begingroup$ OK. That's what I am doing right now! $\endgroup$
    – Johannes
    Feb 11, 2013 at 12:30

3 Answers 3

3
$\begingroup$

This question is rather old and the OP seems to be not among us anymore, but still:

These things are of course found in Bourbaki's Éléments de mathématique, especially in E.III.1.9 and A.VI.1.8.

$\endgroup$
3
$\begingroup$

As far as the standard order on the real line is concerned, most of the above properties are (at least) stated in

Stanisław Łojasiewicz, An introduction to the theory of real functions. With contributions by M. Kosiek, W. Mlak and Z. Opial. Third edition. Translated from the Polish by G. H. Lawden. Translation edited by A. V. Ferreira. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1988. x+230 pp. ISBN: 0-471-91414-2, MR0952856, Zbl 0653.26001.

Analogous properties of sup are proved there; those for inf may be left as exercises. I do not have a copy at hand right now to give an accurate account.

$\endgroup$
0
$\begingroup$

Egbert Harzheim. Ordered Sets. Springer, 2005

$\endgroup$
1
  • $\begingroup$ Thanks, I already tried that. Did not find any of the above theorems. Are they in there? $\endgroup$
    – Johannes
    Feb 11, 2013 at 14:29

Not the answer you're looking for? Browse other questions tagged or ask your own question.