Security Lattice Construction
The way to view the lattice is by taking the two pieces one at a time. Firstly, you have the lattice:
$$
\begin{array}{c}
Clean \\
| \\
Dirty
\end{array}
$$
to capture that Clean data has higher integrity than Dirty data.

Similarly, 
$$
\begin{array}{c}
DataEntry \\
| \\
Website
\end{array}
$$
captures that DataEntry data has higher integrity than Website data.

These can be combined (using the product of two lattices) into the following,
which captures all combinations and the relationship between them:
$$
\begin{array}{ccccc}
&amp;&amp; (Clean, DataEntry)  \\
&amp; / &amp;&amp; \backslash~~~~~~~~~~~~~~~~~~~~~~~~~ \\
 (Dirty, DataEntry)  &amp;&amp;&amp;  (Clean,Website)  \\
&amp;\backslash &amp;&amp;/~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
&amp;&amp;     (Dirty,Website)
\end{array}
$$
The two middle elements are incomparable.

The paper Lattice-based Access Control Models by Ravi Sandhu gives a good overview of the application of lattices in access control models, including the Biba model.

If you want to learn the general theory about lattices, then Introduction to Lattices and Order by B. A. Davey and H. A. Priestley is a very good starting point.