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:

   1. $\mathrm{\neg }$ satisfies the de Morgan’s laws; this means that:

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

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

   2. $\mathrm{\neg }0=1$ and $\mathrm{\neg }1=0$

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 $0$.