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\begin{document}

\begin{center}
{{\Large\bf ALGEBRA 1}\\
\bigskip

{\Large\bf HOMEWORK ASSIGNMENT III}\\

\bigskip

(Turn in underlined problems.)}

\bigskip
Read Sections 1.6, 2.1, 2.3--3.1
\end{center}

\vspace{.5in}

\noindent
{\bf 
\begin{tabbing}
\underline{\large\bf SECTION}\makebox[2in]{}\=\underline{\large\bf PROBLEMS}\\

\bigskip
\noindent
1.6 \>  \underline{7}, 9\\

\noindent
2.1 \> 6, 9, 12, 15\\

\noindent
2.4 \> \underline{17}, 18 

\end{tabbing}
}
\vspace{.5in}

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

{\bf 

These additional problems concern the problem of algebraizing
semilattices and lattices.

\smallskip

\noindent
Definition. A {\bf \em meet semilattice} is a partially ordered
set $\lb P; \leq \rb$ in which each pair of elements
has a greatest lower bound. (The element $d$ is the 
{\bf\em greatest lower bound} or {\bf\em meet} of $a$ and $b$ if 
\begin{enumerate}
\item[(i)] $d\leq a$, $d\leq b$, and
\item[(ii)] if $c\leq a$ and $c\leq b$, then $c\leq d$.
\end{enumerate}
We write $d = a\wedge b$ to indicate that $d$ is the 
meet of $a$ and $b$.
The {\bf\em least upper bound} or {\bf\em join} of
$a$ and $b$ is defined dually, and indicated by $d = a\vee b$.)

\bigskip

\underline{1.} Algebraize the theory of meet semilattices by showing that
if $\m p = \lb P;\leq\rb$ is a meet semilattice, then
\begin{enumerate}
\item[(a)] $\m p$ satisfies
\begin{enumerate}
\item[(i)] $x\wedge x = x$ (idempotence), 
\item[(ii)] $x\wedge y = y\wedge x$ (commutativity), and
\item[(iii)] $x\wedge (y\wedge z) = (x\wedge y)\wedge z$
(associativity).
\end{enumerate}
\item[(b)] If $\m p = \lb P; \wedge\rb$ is an algebra
of type $\lb 2\rb$ satisfying (a)(i)--(iii), and 
$$\leq = \{(a,b)\in P^2\;|\;a = a\wedge b\},$$
then $\leq$ is a partial order on $P$ and 
$a\wedge b$ is the greatest lower bound of $a$ and $b$
with respect to this partial order.
\item[(c)] The correspondences $\lb P; \leq\rb \mapsto 
\lb P; \wedge\rb$ and $\lb P; \wedge\rb \mapsto 
\lb P; \leq\rb$ from parts (a) and (b) are inverses
of one another. (The word ``semilattice'' is used
for either type of structure.)
\end{enumerate}

\bigskip

[Note: If the previous problem had been phrased in terms of
join instead of meet, then we would have defined
the order dually: $\leq = \{(a,b)\in P^2\;|\;b = a\vee b\}.$]

\bigskip

\noindent
Definition. A {\bf \em lattice} is a partially ordered
set $\lb P; \leq \rb$ in which each pair of elements
has a meet and a join.

\bigskip

\underline{2.} 
Algebraize the theory of lattices by showing that
if $\m p = \lb P;\leq\rb$ is a lattice, then 
\begin{enumerate}
\item[(a)] meet and join are idempotent, commutative
and associative operations on $P$ that are related by the
absorption laws: $a\wedge (a\vee b) = a$ and
$a\vee (a\wedge b) = a$.
\item[(b)] If $\m p = \lb P; \vee, \wedge\rb$ is an algebra
of type $\lb 2, 2\rb$ whose operations are idempotent,
commutative, associative, and satisfy the absorption
laws, then the relations
$$\leq = \{(a,b)\in P^2\;|\;a = a\wedge b\}$$
and 
$$\sqsubseteq = \{(a,b)\in P^2\;|\;b = a\vee b\}$$
are equal and $\lb P; \leq \rb$ is a lattice
with meet $=\wedge$ and join $=\vee$.
\end{enumerate}

\bigskip

3.
A {\bf \em power set lattice} is a lattice 
$\m p = \lb {\mathcal P}(X); \cup, \cap\rb$ whose universe
is the set of subsets some set $X$ and whose
operations are union and intersection.
Algebraize these objects by showing that 
\begin{enumerate}
\item[(a)] $\m p$ is a lattice satisfying the 
distributive law: $A\cap (B\cup C) = 
(A\cap B)\cup(A\cap C)$, and
\item[(b)] any lattice $\m p = \lb P; \vee,\wedge\rb$
satisfying the distributive law is embeddable 
in $\lb {\mathcal P}(X); \cup,\cap\rb$ for some set $X$.
\end{enumerate}
(Hints for (b): Let ${\bf 2} = \lb \{0,1\}, \vee,\wedge\rb$
be the 2-element lattice ordered by $0 < 1$.
Choose $X = \{\varphi\;|\;\mbox{\rm 
$\varphi: \m p\rightarrow {\bf 2}$ is a homomorphism.}\}$. 
Then define
$$\Phi: \m p\rightarrow \lb {\mathcal P}(X); \cup, \cap\rb$$
by $\Phi(a) = \{\varphi\;|\;\varphi(a) = 1\}$.
Prove or take my word for it that for any 
$a\neq b\in P$ there is a $\varphi\in X$
such that $\varphi(a)\neq\varphi(b)$, and hence $\Phi$
is 1-1. The argument for this requires some form
of Zorn's lemma. 
Now show that $\Phi$ preserves meet and join.)

\bigskip

4. Notice that any power set lattice is isomorphic to its
dual (which is the lattice obtained by reversing the order,
and hence interchanging meet and join). A specific
isomorphism of ${\mathcal P}(X)$ with 
${\mathcal P}(X)^{\partial}$ is the function
$A\mapsto X-A$ (complementation). Hence, if any
sentence is true in all power set lattices, then
its dual is also true in all power set lattices.

The result of the previous problem implies that 
the universally quantified laws true in all power
set lattices are exactly the consequences of 
the lattice laws and the distributive law.
The lattice laws are self-dual, but the distributive
law is not. Consequently, we deduce the non-obvious
fact that {\bf\em in the presence
of the laws of lattice theory, one distributive
law 
$$x\wedge(y\vee z) = (x\wedge y)\vee(x\wedge z)$$
implies its dual
$$x\vee(y\wedge z) = (x\vee y)\wedge(x\vee z).$$}
Give an equational deduction of this fact.

}
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