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

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

\vspace{.5in}

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

{\bf 

1.
Suppose that $\langle A;R\rangle$ is a structure where
$A$ is a set and $R$ is a binary relation on $A$. Show that 
the following are equivalent:
\begin{enumerate}
\item[(i)]
$\langle A;R\rangle$ is a partially ordered set.
\item[(ii)] There is a structure 
$\langle A';R'\rangle$ isomorphic to 
$\langle A;R\rangle$ where each element of $A'$
is a set and $(X,Y)\in R'$ if and only if 
$X$ is a subset of $Y$.
\end{enumerate}

\bigskip

\underline{2.} 
Suppose that a finite ordered set $\langle P;\leq\rangle$
is a semilattice with a top element.
Show that it is a lattice. Give an example
to show that this statement is not true for 
infinite ordered sets.

\bigskip

3. Draw the lattice of divisors of $72$.
\bigskip

\underline{4.}
Show that a lattice that satisfies one distributive law:
\[x\wedge (y\vee z) = (x\wedge y)\vee (x\wedge z)\]
also satisfies the other one:
\[x\vee (y\wedge z) = (x\vee y)\wedge (x\vee z).\]
Thus, the dual of a distributive lattice is distributive.

\bigskip

\underline{5.} Let ${\bf L}$ be a lattice. For a fixed $a\in L$
define a function 
\[J_a(x):L\rightarrow L: x\mapsto a\vee x.\]
Show that if ${\bf L}$ is distributive, then 
$J_a(x\vee y) = J_a(x)\vee J_a(y)$ and
$J_a(x\wedge y) = J_a(x)\wedge J_a(y)$.

\bigskip

6. Explain why the lattice of subsets of a set 
(ordered by inclusion) is a distributive lattice.

\bigskip

\underline{7.} Show that the lattice of subspaces of a 
vector space is modular.

\bigskip

\underline{8.} Show that the lattice of equivalence relations
on $X = \{1, 2, 3, 4\}$ is not modular.

}
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