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sav08:lattices [2009/03/31 23:41] vkuncak |
sav08:lattices [2010/04/08 11:00] vkuncak |
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This does not imply there are lub and glb for infinite sets. | 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 $\sqcap S$ | **Definition:** A **complete** lattice is a lattice where every set $S$ of elemenbts has lub, denoted $\sqcup S$, and glb, denoted $\sqcap S$ | ||
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===== References ===== | ===== 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/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]]. | * [[http://bigcheese.math.sc.edu/~mcnulty/alglatvar/burrissanka.pdf|Chapter I of a Course in Universal Algebra]]. |