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complete lattice (Definition)

Complete lattices

A complete lattice is a poset $P$ such that every subset of $P$ has both a supremum and an infimum in $P$.

For a complete lattice $L$, the supremum of $L$ is denoted by $1$, and the infimum of $L$ is denoted by $0$. Thus $L$ is a bounded lattice, with $1$ as its greatest element and $0$ as its least element. Moreover, $1$ is the infimum of the empty set, and $0$ is the supremum of the empty set.

Generalizations

A countably complete lattice is a poset $P$ such that every countable subset of $P$ has both a supremum and an infimum in $P$.

Let $\kappa$ be an infinite cardinal. A $\kappa$-complete lattice is a lattice $L$ such that for every subset $A\subseteq L$ with $\vert A\vert\le \kappa$, both $\bigvee A$ and $\bigwedge A$ exist. (Note that an $\aleph_0$-complete lattice is the same as a countably complete lattice.)

Every complete lattice is a $\kappa$-complete lattice for every infinite cardinal $\kappa$, and in particular is a countably complete lattice. Every countably complete lattice is a bounded lattice.



"complete lattice" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: Tarski-Knaster theorem, complete lattice homomorphism, domain, complete semilattice, complete Boolean algebra, arbitrary join

Also defines:  countably complete lattice, countably-complete lattice, $\kappa$-complete, $\kappa$-complete lattice

Attachments:
criteria for a poset to be a complete lattice (Theorem) by CWoo
bounded complete (Definition) by CWoo
intersection structure (Definition) by CWoo
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Cross-references: lattice, cardinal, infinite, countable, empty set, least element, greatest element, bounded lattice, infimum, supremum, subset, poset
There are 38 references to this entry.

This is version 7 of complete lattice, born on 2002-08-17, modified 2008-02-20.
Object id is 3304, canonical name is CompleteLattice.
Accessed 5286 times total.

Classification:
AMS MSC: 06B23 (Order, lattices, ordered algebraic structures :: Lattices :: Complete lattices, completions)
 03G10 (Mathematical logic and foundations :: Algebraic logic :: Lattices and related structures)
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Countably complete semilattice by porton on 2007-12-25 11:25:32
What's about a special name for the poset for which every countable subset has supremum? Could it be called "countably complete join-semilattice"? Or does it have any other special name?
--
Victor Porton - http://www.mathematics21.org
* Algebraic General Topology and Math Synthesis
* 21 Century Math Method (post axiomatic math logic)
* Category Theory - new concepts
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