LatticEs

A 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.