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\begin{document}

\maketitle

\begin{center}
{{\Large\bf ALGEBRA I}\\
\bigskip

{\Large\bf HOMEWORK ASSIGNMENT II}\\
\bigskip
{\bf (Turn in underlined problems.)}}
\end{center}

\vspace{.5in}

\noindent
\underline{\large\bf PROBLEMS}
\vspace{.25in}

{\bf 

\underline{1.}
Show that in a modular lattice the meet of 
\[(y\vee(x\wedge(y\vee z)))\wedge(z\vee((x\wedge y)\vee (x\wedge z)))\]
and 
\[(x\wedge(y\vee z))\vee
((y\vee((x\wedge y)\vee (x\wedge z)))
\wedge(z\vee((x\wedge y)\vee (x\wedge z))))\]
is 
\[(y\vee((x\wedge y)\vee (x\wedge z)))
\wedge(z\vee((x\wedge y)\vee (x\wedge z))).\]

\bigskip

\underline{2.} Equivalence relations $\theta$ and $\psi$ on a set $X$
{\it permute} if $\theta\circ \psi = 
\psi\circ \theta$. Show that if 
$\theta$ and $\psi$ permute, then 
$\theta\vee \psi = \theta\circ\psi$.


\bigskip

\underline{3.}
Show that if $L$ is a lattice of permuting equivalence 
relations on a set $X$, then $L$ is modular.


\bigskip

4. Equivalence relations $\theta$ and $\psi$ on a set $X$
{\it 3-permute} if $\theta\circ \psi \circ \theta= 
\psi\circ \theta \circ \psi$. 
Show that if $L$ is a lattice of 3-permuting equivalence 
relations on a set $X$, then $L$ is modular.
(If you replace 3-permuting
equivalence relations 
by ``4-permuting'' equivalence
relations, then you may get a nonmodular lattice.)

\bigskip

\underline{5.} Show that the lattice of positive
integers ordered by divisibility is a distributive
lattice. (This is the reason that the Generalized Chinese
Remainder Theorem is true.) 

\bigskip

\underline{6.} 
Show that a distributive lattice of finite height 
is finite.
}
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