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Wikipedia / Bounded lattice

In mathematics, and in particular in order theory, a bounded lattice is a lattice that has a least element and a greatest element, respectively.

Bounded lattices are of considerable importance because many algebraic structures are bounded lattices, including complete lattices, Heyting algebras, Boolean algebrass, and others.

Definition

A bounded lattice can be defined in two equivalent ways: via an order relation or algebraically. These two definitions can be shown to be equivalent.

Order-theoretic definition

be a partially ordered set is called a bounded lattice if and only if:

is a lattice with respect to the order relation:

  1. , there exists an infimum.

  2. , there exists a supremum.

is a bounded poset:

  1. .

  2. .

Algebraic definition

be a set equipped with two binary operations is called a bounded lattice if and only if the following conditions hold:

:

  1. Associativity.

  2. Commutativity.

  3. Idempotence.

  4. .

are identity elements, respectively:

  1. .

  2. .

Properties

  • .

  • .

Bounding a lattice

can be order-embedded.

, a collection of subsets denotes the principal lower set is an order embedding.

The Dedekind–MacNeille completion proves a much stronger statement: every partially ordered set (not necessarily a lattice) can be embedded into a complete lattice (which is necessarily bounded).

Complemented lattice

be a bounded lattice. It is called a complemented lattice.

is called a complement. In contrast to a Boolean algebra, a complemented lattice may have more than one complement for a given element. Intuitively, a complement can be thought of as a negation of the element.

Examples

  • Every finite partially ordered set that is a lattice is a complete lattice, and hence a bounded lattice.

    Every closed interval in the real line is a bounded lattice.

    be the collection of all vector subspaces.

    serve as global minimum and maximum.

    be the collection of all convex and closed polygons, ordered by inclusion. This is a lattice because the intersection of two closed convex polygons is again a closed convex polygon, and the closure of the convex hull as maximum. However, it is not complete: the family of all closed convex polygons contained in the unit disk has no supremum in this lattice, since their union is a circle, which is not a polygon.

References