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A 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. TotalOrderedSets and BooleanAlgebrae are two important types of lattice.

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Last edited January 29, 2001 9:00 am by JoshuaGrosse (diff)