Lattices and Categories

Abstract

The subsets of a set permit operations quite similar to those performed on numbers. If for the moment we denote the union of two subsets A, B by A +B and their intersection by AB, a notation that will not be used later (despite some historical precedents), then we have laws like AB = BA, A(B + C) = AB + AC, similar to the familiar laws of arithmetic, as well as new laws such as A + A = A, A + BC = (A + B)(A + C). The algebra formed in this way is called a Boolean algebra, after George Boole who introduced it around the middle of the 19th century, and who made the interesting observation that Boolean algebras could also be used to describe the propositions of logic.