³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980274539³2ip³2129.97.112.43³2data³2newauthor³31³3minor³30³3text³3A lattice is a PartialOrderedSet (X, <=) where any two elements x,y have both a GreatestLowerBound xvy and LeastUpperBound x^y.

The operations v,^ on a lattice form an algebraic structure satisfying the following properties:

    ava=a                   a^a=a                   (idempotency laws)
    avb=bva                 a^b=b^a                 (commutativity laws)
    av(bvc)=(avb)vc         a^(b^c)=(a^b)^c         (associativity laws)
    av(a^b)=a               a^(bvc)=a               (absorption laws)

Moreover, any algebra satisfying the above properties is a lattice, where x<=y is defined to be true if and only if xvy=y (or equivalently, x^y=x).

A lattice which satisfies the following properties is referred to as a boolean algebra:

    There exists some element 0, such that av0=a for all a                 (bounded below)
    There exists some element 1, such that a^1=a for all a                 (bounded above)
    For all a,b,c, (avb)^c=(a^c)v(b^c) and (a^b)vc=(avc)^(bvc)             (distributive laws - these are equivalent)
    For every a, there exists an element ~a such that av~a=1 and a^~a=0    (existence of complements)

Complements are unique within bounded distributive lattices.  Boolean algebrae are very similar to RinGs, except they have complements instead of inverses, and moreover the distributive laws work for each operation over the other.

The subobjects of any given object form a boolean algebra under <= taken to mean "is a subobject of".  In the case of sets, v and ^ become unions and intersections; in the case of groups they become semidirect products and intersections, and so forth.
³3summary³3³2tscreate³2980274539³2keepts³2980402816³2id³21076³2host³2cnts1p43.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980402816³2ip³2129.97.182.64³2data³2text³3A lattice is a PartialOrderedSet (X, <=) where any two elements x,y have both a GreatestLowerBound xvy and LeastUpperBound x^y.

The operations v,^ on a lattice form an algebraic structure satisfying the following properties:

    ava=a                   a^a=a                   (idempotency laws)
    avb=bva                 a^b=b^a                 (commutativity laws)
    av(bvc)=(avb)vc         a^(b^c)=(a^b)^c         (associativity laws)
    av(a^b)=a               a^(bvc)=a               (absorption laws)

Moreover, any algebra satisfying the above properties is a lattice, where x<=y is defined to be true if and only if xvy=y (or equivalently, x^y=x).

One ''very'' important kind of lattice is a BooleanAlgebra.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980274539³2keepts³2980410166³2id³21076³2host³2cnts2p64.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980410166³2ip³2129.97.182.28³2data³2newauthor³30³3minor³31³3text³3A lattice is a set L, together with two binary operations v,^, such that for any a,b,c in L,

    ava=a                   a^a=a                   (idempotency laws)
    avb=bva                 a^b=b^a                 (commutativity laws)
    av(bvc)=(avb)vc         a^(b^c)=(a^b)^c         (associativity laws)
    av(a^b)=a               a^(bvc)=a               (absorption laws)

If avb=b, or equivalently a^b=a, we say that a<=b.  Thus defined, (L,<=) is a PartialOrderedSet, and moreover (L,v,^) is the unique lattice associated therewith.  TotalOrderedSet''s'' and BooleanAlgebra''e'' are two important types of lattice.

³3summary³3*³2tscreate³2980274539³2keepts³2980496461³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980496461³2ip³2129.97.130.42³2data³2text³3A lattice is a set L, together with two binary operations v,^, such that for any a,b,c in L,

    ava=a                   a^a=a                   (idempotency laws)
    avb=bva                 a^b=b^a                 (commutativity laws)
    av(bvc)=(avb)vc         a^(b^c)=(a^b)^c         (associativity laws)
    av(a^b)=a               a^(bvc)=a               (absorption laws)

If avb=b, or equivalently a^b=a, we say that a<=b.  Thus defined, <= forms a PartialOrder on L, and moreover (L,v,^) is the unique lattice associated therewith.  TotalOrderedSet''s'' and BooleanAlgebra''e'' are two important types of lattice.

³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980274539³2keepts³2980496639³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³24