Ockham algebra

A latticeMathworldPlanetmath L is called an Ockham algebra if

  1. 1.
  2. 2.

    L is bounded, with 0 as the bottom and 1 as the top

  3. 3.

    there is a unary operator ¬ on L with the following properties:

    1. (a)

      ¬ satisfies the de Morgan’s laws; this means that:

      • *

        ¬(ab)=¬a¬b and

      • *


    2. (b)

      ¬0=1 and ¬1=0

Such a unary operator is an example of a dual endomorphismMathworldPlanetmathPlanetmathPlanetmath. When applied, ¬ interchanges the operationsMathworldPlanetmath of and , and 0 and 1.

An Ockham algebra is a generalizationPlanetmathPlanetmath of a Boolean algebraMathworldPlanetmath, in the sense that ¬ replaces , the complementMathworldPlanetmathPlanetmath operator, on a Boolean algebra.


  • An intermediate concept is that of a De Morgan algebra, which is an Ockham algebra with the additional requirement that ¬(¬a)=a.

  • In the categoryMathworldPlanetmath of Ockham algebras, the morphism between any two objects is a {0,1}-lattice homomorphismMathworldPlanetmath (http://planetmath.org/LatticeHomomorphism) f that preserves ¬: f(¬a)=¬f(a). In fact, f(0)=f(¬1)=¬f(1)=¬1=0, so that it is safe to drop the assumptionPlanetmathPlanetmath that f preserves 0.


  • 1 T.S. Blyth, J.C. Varlet, Ockham Algebras, Oxford University Press, (1994).
  • 2 T.S. Blyth, Lattices and Ordered Algebraic StructuresPlanetmathPlanetmath, Springer, New York (2005).
Title Ockham algebra
Canonical name OckhamAlgebra
Date of creation 2013-03-22 17:08:34
Last modified on 2013-03-22 17:08:34
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 06D30