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sav08:lattices [2009/03/31 23:44] vkuncak |
sav08:lattices [2011/05/03 16:46] vkuncak |
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**Definition:** A lattice is a [[:Partial order]] in which every two-element set has a least upper bound and a greatest lower bound. | **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!** ++ \\ | **Proof:** is by ++| **induction!** ++ \\ | ||
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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]]. |