Ockham algebra
A lattice^{} $L$ is called an Ockham algebra if

1.
$L$ is distributive^{}

2.
$L$ is bounded, with $0$ as the bottom and $1$ as the top

3.
there is a unary operator $\mathrm{\neg}$ on $L$ with the following properties:

(a)
$\mathrm{\neg}$ satisfies the de Morgan’s laws; this means that:

*
$\mathrm{\neg}(a\vee b)=\mathrm{\neg}a\wedge \mathrm{\neg}b$ and

*
$\mathrm{\neg}(a\wedge b)=\mathrm{\neg}a\vee \mathrm{\neg}b$

*

(b)
$\mathrm{\neg}0=1$ and $\mathrm{\neg}1=0$

(a)
Such a unary operator is an example of a dual endomorphism^{}. When applied, $\mathrm{\neg}$ interchanges the operations^{} of $\vee $ and $\wedge $, and $0$ and $1$.
An Ockham algebra is a generalization^{} of a Boolean algebra^{}, in the sense that $\mathrm{\neg}$ replaces ${}^{\prime}$, the complement^{} operator, on a Boolean algebra.
Remarks.

•
An intermediate concept is that of a De Morgan algebra, which is an Ockham algebra with the additional requirement that $\mathrm{\neg}(\mathrm{\neg}a)=a$.

•
In the category^{} of Ockham algebras, the morphism between any two objects is a $\{0,1\}$lattice homomorphism^{} (http://planetmath.org/LatticeHomomorphism) $f$ that preserves $\mathrm{\neg}$: $f(\mathrm{\neg}a)=\mathrm{\neg}f(a)$. In fact, $f(0)=f(\mathrm{\neg}1)=\mathrm{\neg}f(1)=\mathrm{\neg}1=0$, so that it is safe to drop the assumption^{} that $f$ preserves $0$.
References
 1 T.S. Blyth, J.C. Varlet, Ockham Algebras, Oxford University Press, (1994).
 2 T.S. Blyth, Lattices and Ordered Algebraic Structures^{}, Springer, New York (2005).
Title  Ockham algebra 

Canonical name  OckhamAlgebra 
Date of creation  20130322 17:08:34 
Last modified on  20130322 17:08:34 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06D30 