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 $\neg$ on $L$ with the following properties:
   1. $\neg$ 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. $\neg 0=1$ and $\neg 1=0$

An Ockham algebra is a generalization of a Boolean algebra, in the sense that $\neg$ replaces $'$ , 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 $\lbrace 0,1\rbrace$ -lattice homomorphism $f$ 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 $0$ .

Bibliography

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).