Boolean Algebra
A mathematical object which is similar to a Boolean Ring, but which uses the meet and join operators instead of theusual addition and multiplication operators. A Boolean algebra is a set 
of elements 
, 
, ... with BinaryOperators 
and 
such that
- 1a. If and are in the set , then
is in the set . - 1b. If and are in the set , then
is in the set . - 2a. There is an element
(zero) such that 
for every element . - 2b. There is an element
(unity) such that 
for every element . - 3a.
- 3b.
- 4a.
- 4b.
- 5. For every element there is an element
such that 
and 
. - 6. There are at least two distinct elements in the set .
In more modern terms, a Boolean algebra is a Set of elements , , ... with the followingproperties:
- 1. has two binary operations,
(Wedge) and 
(Vee), which satisfy theIdempotent laws
the Commutative laws
and the Associative laws
- 2. The operations satisfy the Absorption Law
- 3. The operations are mutually distributive
- 4. contains universal bounds
which satisfy
- 5. has a unary operation
of complementation which obeys the laws
form a Boolean algebra.Huntington (1933a, b) presented the following basis for Boolean algebra,
- 1. Commutativity.
. - 2. Associativity.
. - 3. Huntington Equation.
.
The Algebra defined by commutativity, associativity, and the Robbins Equation is called RobbinsAlgebra. Computer theorem proving demonstrated that every Robbins Algebra satisfies the second WinklerCondition, from which it follows immediately that all Robbins Algebras areBoolean.
References
Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1986. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 3rd ed. New York: Macmillian, p. 317, 1965. Halmos, P. Lectures on Boolean Algebras. Princeton, NJ: Van Nostrand, 1963. Huntington, E. V. ``New Sets of Independent Postulates for the Algebra of Logic.'' Trans. Amer. Math. Soc. 35, 274-304, 1933a. Huntington, E. V. ``Boolean Algebras: A Correction.'' Trans. Amer. Math. Soc. 35, 557-558, 1933. McCune, W. ``Robbins Algebras are Boolean.'' http://www-unix.mcs.anl.gov/~mccune/papers/robbins/.
- Everett's Formula
- Eversion
- Evolute
- Exact Covering System
- Exact Differential
- Exactly One
- Exactly When
- Exact Period
- Exact Trilinear Coordinates
- Excenter
- Excenter-Excenter Circle
- Excentral Triangle
- Excess
- Excess Coefficient
- Excessive Number
- Excircle
- Excision Axiom
- Excluded Middle Law
- Excludent
- Excludent Factorization Method
- Exclusive Or
- Exeter Point
- Exhaustion Method
- Existence
- Existential Closure