Differences

This shows you the differences between two versions of the page.

--- sav08:lattices [2009/03/31 23:41]
vkuncak
+++ sav08:lattices [2011/05/03 16:46]
vkuncak
@@ Line 3: / Line 3: @@
 **Definition:** A lattice is a [[:Partial order]] in which every two-element set has a least upper bound and a greatest lower bound.
-**Lemma:** In a lattice every non-empty set has a lub ($\sqcup$) and glb ($\sqcap$).
+**Lemma:** In a lattice every non-empty finite set has a lub ($\sqcup$) and glb ($\sqcap$).
 **Proof:** is by ++| **induction!** ++ \\
@@ Line 20: / Line 20: @@
 This does not imply there are lub and glb for infinite sets.
-**Example:** The set of all rational numbers. Take $[0,1)$, or $\{ x \mid 0 \le x \land x^2 < 2 \}$ have no lub.
+**Example:** In the oder $([0,1),\le)$ with standard ordering on reals is a lattice, the entire set has no lub.
+The set of all rationals of interval $[0,10]$ is a lattice, but the set $\{ x \mid 0 \le x \land x^2 < 2 \}$ has no lub.
 **Definition:** A **complete** lattice is a lattice where every set $S$ of elemenbts has lub, denoted $\sqcup S$, and glb, denoted $\sqcap S$
@@ Line 49: / Line 50: @@
 ===== References =====
-  * [[wk>Lattice (order)]]
+  * [[wp>Lattice (order)]]
   * [[http://bigcheese.math.sc.edu/~mcnulty/alglatvar/lat0.pdf|lecture notes by J.B. Nation]] or
   * [[http://bigcheese.math.sc.edu/~mcnulty/alglatvar/burrissanka.pdf|Chapter I of a Course in Universal Algebra]].