In mathematics, and in particular in order theory, a bounded lattice is a lattice that has a least element and a greatest element, usually denoted by and , respectively.
Bounded lattices are of considerable importance because many algebraic structures are bounded lattices, including complete lattices, Heyting algebras, Boolean algebras, 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
Let be a partially ordered set. Then is called a bounded lattice if and only if:
is a lattice with respect to the order relation: For every pair , there exists an infimum. For every pair , there exists a supremum. is a bounded poset: There exists such that for every , . This element is unique and is denoted by . There exists such that for every , . This element is unique and is denoted by .
Algebraic definition
Let be a set equipped with two binary operations and , and two distinguished elements . Then is called a bounded lattice if and only if the following conditions hold:
is a lattice with respect to and : Associativity: for all , and . Commutativity: for all , and . Idempotence: for all , and . Absorption: for all , and . and are identity elements for and , respectively: For all , . For all , . Properties
In a bounded lattice , for every , one has . In a bounded lattice , for every , one has .
Bounding a lattice
Let be an arbitrary lattice. One may ask whether there exists a bounded lattice into which can be order-embedded.
Define , a collection of subsets of , where for each , denotes the principal lower set generated by . It can be shown that , ordered by inclusion , is a bounded lattice. Define a function by . One can prove that 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
Let be a bounded lattice. It is called a complemented lattice if and only if for every there exists such that and .
In this case, is called a complement of . 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.
Let be the collection of all vector subspaces of , ordered by inclusion. This is a bounded lattice with minimum and maximum .
Let be the set of all continuous functions from to the closed interval , ordered pointwise: if and only if for all . Then is a lattice, since for any two functions one may take their pointwise minimum and maximum, which are again continuous. However, for an infinite family of continuous functions, this construction need not yield a continuous function. Hence, this lattice is not complete. It is bounded, since the constant functions and serve as global minimum and maximum.
Let be the collection of all convex and closed polygons in , together with the empty set and the whole space , 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 of their union is also a closed convex polygon. It is bounded, with as minimum and 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.
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