Definitions

1. A complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).
2. A complete lattice is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.
3. Any complete lattice (also see below) is a (rather specific) bounded lattice.
4. Every complete lattice is also a bounded lattice, which is to say that is has a greatest and least element. (Web site)

Algebraic Lattice

1. An algebraic lattice is a complete lattice that is algebraic as a poset.

Complete Semilattice

1. Every poset that is a complete semilattice is also a complete lattice. (Web site)

Particular

1. In particular, every complete lattice is a bounded lattice. (Web site)

Finite Set

1. Moreover, every lattice with a finite set is a complete lattice.

Convex Sets

1. The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice. (Web site)

Fixed Points

1. An example is the Knaster-Tarski theorem, which states that the set of fixed points of a monotone function on a complete lattice is again a complete lattice.

Complete Boolean Algebra

1. A complete Boolean algebra is a Boolean algebra that is a complete lattice. (Web site)
2. Notable special usages of the term include the concepts of complete Boolean algebra, complete lattice, and complete partial order (cpo).

Complete Heyting Algebra

1. In fact, the set of open set s provides a classical example of a complete lattice, more precisely a complete Heyting algebra (or " frame " or " locale "). (Web site)

Set

1. A totally ordered set (with its order topology) which is a complete lattice is compact. (Web site)
2. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice. (Web site)
3. The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete lattice. (Web site)

Partial Order

1. The relation of "being-finer-than" is a partial order on the set of all partitions of the set X, and indeed even a complete lattice.

Subset

1. Let X be a complete lattice, then a function f: X → X is continuous if, for each subset Y of X, we have sup f(Y) = f(sup Y).

Closure Operator

1. If P is a complete lattice, then a subset A of P is the set of closed elements for some closure operator on P if and only if A is a Moore family on P, i.e. (Web site)
2. In summary, one can say that every complete lattice is isomorphic to the image of a closure operator on a powerset lattice. (Web site)

Heyting Algebra

1. A complete Heyting algebra is a Heyting algebra that is a complete lattice. (Web site)

Lattice

1. The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (Web site)
2. If the order is a meet-semilattice (or even a lattice), then this construction (including the empty filter) actually yields a complete lattice.
3. Lattice of topologies The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete lattice.

Poset

1. A poset, considered as a small category, is complete (and cocomplete) if and only if it is a complete lattice. (Web site)
2. Completeness A poset is called a complete lattice if all its subsets have both a join and a meet.
3. A continuous lattice is a complete lattice that is continuous as a poset.

Infimum

1. In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet).
2. This order has the desirable property that every subset has a supremum and an infimum: it is a complete lattice. (Web site)

Complete Lattice

1. It is a totally ordered set and a complete lattice (every subset of the unit interval has a supremum and an infimum). (Web site)
2. A lattice in which every subset (including infinite ones) has a supremum and an infimum is called a complete lattice.
3. As explained in the article on completeness (order theory), any poset for which either all suprema or all infima exist is already a complete lattice.

Categories

1. Infimum
2. Poset
3. Supremum
4. Lattice
5. Heyting Algebra
6. Books about "Complete Lattice" in Amazon.com
   

Continue: More Keywords - - - - - - - - - -