# Ockham algebra

A lattice $L$ is called an Ockham algebra if

1. 1.

$L$ is distributive

2. 2.

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

3. 3.

there is a unary operator $\neg$ on $L$ with the following properties:

1. (a)

$\neg$ satisfies the de Morgan’s laws; this means that:

• *

$\neg(a\vee b)=\neg a\wedge\neg b$ and

• *

$\neg(a\wedge b)=\neg a\vee\neg b$

2. (b)

$\neg 0=1$ and $\neg 1=0$

Such a unary operator is an example of a dual endomorphism. When applied, $\neg$ 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 $\neg$ 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 $\neg(\neg a)=a$.

• 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 $\neg$: $f(\neg a)=\neg f(a)$. In fact, $f(0)=f(\neg 1)=\neg f(1)=\neg 1=0$, 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 OckhamAlgebra 2013-03-22 17:08:34 2013-03-22 17:08:34 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 06D30