## You are here

Homecomplete lattice

## Primary tabs

# complete lattice

# 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 $|A|\leq\kappa$, 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.

## Mathematics Subject Classification

06B23*no label found*03G10

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## Countably complete semilattice

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

## Re: Countably complete semilattice

porton writes:

> Could it be called "countably complete join-semilattice"?

Sounds good to me. But it's a rather obscure concept, so it's hard to find any examples of this term. I did find one example of "countably complete upper semilattice".

## Re: Countably complete semilattice

> porton writes:

>

> > Could it be called "countably complete join-semilattice"?

>

> Sounds good to me. But it's a rather obscure concept, so

> it's hard to find any examples of this term. I did find one

> example of "countably complete upper semilattice".

Googling of "countably complete upper semilattice" | "countably complete upper lattice" finds no results. Are there any standard term for countably complete semilattices?

--

Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis

* 21 Century Math Method (post axiomatic math logic)

* Category Theory - new concepts

## Re: Countably complete semilattice

Porton writes:

> Googling of "countably complete upper semilattice" |

> "countably complete upper lattice" finds no results.

Googling for "countably complete upper semilattice" gets one result if use Google Books ( http://books.google.com ) rather than the web search.

## Re: Countably complete semilattice

> Porton writes:

>

> > Googling of "countably complete upper semilattice" |

> > "countably complete upper lattice" finds no results.

>

> Googling for "countably complete upper semilattice" gets one

> result if use Google Books ( http://books.google.com )

> rather than the web search.

No on Google Books also zero search results.

Which book to you refer?

Finally, are there such the term, or maybe we should invent it?

--

Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis

* 21 Century Math Method (post axiomatic math logic)

* Category Theory - new concepts

## Re: Countably complete semilattice

porton writes:

> No on Google Books also zero search results.

Well, it works for me.

> Which book to you refer?

Recent Trends in Algebraic Development Techniques

14th International Workshop, WADT'99

Chateau de Bonas, France, September 1999

Selected Papers

editors: Didier Bert, Christine Choppy, Peter Mosses

Lecture Notes in Computer Science 1827

Springer

It's on page 206, in a paper by Yoshiki Kinoshita and John Power.