| Version 3 |
Version 2 |
| A lattice $L$ is called an \emph{Ockham algebra} if |
A lattice $L$ is called an \emph{Ockham algebra} if |
| \begin{enumerate} |
\begin{enumerate} |
| \item $L$ is distributive |
\item $L$ is distributive |
| \item $L$ is bounded, with $0$ as the bottom and $1$ as the top |
\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: |
\item there is a unary operator $\neg$ on $L$ with the following properties: |
| \begin{enumerate} |
\begin{enumerate} |
|
\item $\neg$ satisfies the de Morgan's laws; this means that:
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\item $\neg$ satisfies the de Morgan's laws, this means that
|
| \begin{itemize} |
\begin{itemize} |
|
\item $\neg (a\vee b)=\neg a\wedge \neg b$ and
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\item $\neg (a\vee b)=\neg a\wedge \neg b$, and
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| \item $\neg (a\wedge b)=\neg a\vee \neg b$ |
\item $\neg (a\wedge b)=\neg a\vee \neg b$.
|
| \end{itemize} |
\end{itemize} |
| \item $\neg 0=1$ and $\neg 1=0$ |
\item $\neg 0=1$ and $\neg 1=0$.
|
| \end{enumerate} |
\end{enumerate} |
| \end{enumerate} |
\end{enumerate} |
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| An Ockham algebra is a generalization of a Boolean algebra, in the sense that $\neg$ replaces $'$, the complement operator, on a Boolean algebra. |
An Ockham algebra is a generalization of a Boolean algebra, in the sense that $\neg$ replaces $'$, the complement operator, on a Boolean algebra. |
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| An intermediate concept is that of a de Morgan algebra, which is an Ockham algebra with the additional requirement that $\neg (\neg a)=a$. |
An intermediate concept is that of a de Morgan algebra, which is an Ockham algebra with the additional requirement that $\neg (\neg a)=a$. |
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| \begin{thebibliography}{8} |
\begin{thebibliography}{8} |
| \bibitem{bv} T.S. Blyth, J.C. Varlet, {\em Ockham Algebras}, Oxford University Press, (1994). |
\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). |
\bibitem{tsb} T.S. Blyth, {\em Lattices and Ordered Algebraic Structures}, Springer, New York (2005). |
| \end{thebibliography} |
\end{thebibliography} |
