BooleanAlgebra

A boolean algebra is a lattice (see LatticEs) which satisfies the following properties:

    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)                                     (distributive law)
    For all a, there exists an element ~a such that av~a=1 and a^~a=0      (existence of complements)

Complements are guaranteed to be unique within bounded distributive lattices. Note the definition of Boolean algebrae is very similar to that of RinGs, except elements have complements instead of inverses. Moreover the distributive law can be shown to hold both ways, i.e. (a^b)vc=(avc)^(bvc).

A particularly important boolean algebra is given by the lattice {0,1} with ordering =>, defined so that 0=>1. This is because this represents ordinary logical truth values: 0 is "false", 1 is "true", avb is "a or b", a^b is "a and b", and a=>b is "a implies b".

As an immediate consequence of the above, the MathematicalRelations over a set form a boolean algebra under these operations. So, then, do the subsets of any given set under inclusions, and in fact the subobjects of any given mathematical object typically form a boolean algebra under inclusions.