Lattice

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.