Ockham algebra
OckhamAlgebra
2007-05-23 15:36:34
2007-05-24 13:32:15
Definition
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A lattice $L$ is called an \emph{Ockham algebra} if
\begin{enumerate}
\item $L$ is distributive
\item $L$ is bounded, with $0$ as the bottom and $1$ as the top
\item there is a unary operator $\neg$ on $L$ with the following properties:
\begin{enumerate}
\item $\neg$ satisfies the de Morgan's laws; this means that:
\begin{itemize}
\item $\neg (a\vee b)=\neg a\wedge \neg b$ and
\item $\neg (a\wedge b)=\neg a\vee \neg b$
\end{itemize}
\item $\neg 0=1$ and $\neg 1=0$
\end{enumerate}
\end{enumerate}
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 $'$, the complement operator, on a Boolean algebra.
\textbf{Remarks}.
\begin{itemize}
\item
An intermediate concept is that of a De Morgan algebra, which is an Ockham algebra with the additional requirement that $\neg (\neg a)=a$.
\item
In the category of Ockham algebras, the morphism between any two objects is a $\lbrace 0,1\rbrace$-\PMlinkname{lattice homomorphism}{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 $0$.
\end{itemize}
\begin{thebibliography}{8}
\bibitem{bv} T.S. Blyth, J.C. Varlet, {\em Ockham Algebras}, Oxford University Press, (1994).
\bibitem{tsb} T.S. Blyth, {\em Lattices and Ordered Algebraic Structures}, Springer, New York (2005).
\end{thebibliography}