\documentclass[twoside,a4paper,12pt,draft]{article}
\usepackage{amssymb,amsmath,amsthm}
\title{Model-theory homework}
\date{2001, fall}
\author{Math 736---David Pierce}
\pagestyle{myheadings}
 \markboth{Model-theory homework}{Math 736, fall 2001}
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\newtheorem{problem}{Problem}


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\newcommand{\Br}{\mathcal{M}^\mathrm{r}}  % a Boolean ring

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\begin{document}
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\noindent\emph{Some homework for Math 736, Model-Theory, given \today.}

Let $\lang$ be a signature, and let $\str M$ be a structure in
$\Mod(\lang)$.

\begin{problem}
We didn't actually \emph{define} terms of $\lang$; we simply asserted
that terms $t$ exist whose interpretations $t^{\str M}$ have certain
properties.  Prove this assertion.
\end{problem}

\begin{problem}
Prove the lemma that, if $t$ is an $n$-ary term of $\lang$, and $u_0$,
\dots, $u_{n-1}$ are $m$-ary terms of $\lang$, then there is an $m$-ary
term of $\lang$ whose interpretation in ${\str M}$ is the map
\begin{equation*}
\tuple a\mapsto t^{\str M}(u_0^{\str M}(\tuple a),\dots,u_{n-1}^{\str
M}(\tuple a)):M^m\to M.
\end{equation*}
\end{problem}

\begin{problem}
Suppose that a structure in the signature $\{\land,\lor,\lnot,0,1\}$ can
be expanded to a signature containing $+$ in such a way that
  the identities
  $x\lor y  ={x+y}+(x\land y)$ and
  $\lnot x  =x+1$
are satisfied; suppose further that this expansion, reduced to the
signature $\{+,\land,0,1\}$, is a Boolean ring.  Then the original
structure is, by definition, a Boolean algebra.
\begin{enumerate}
  \item
  Prove that the Boolean algebras are precisely those structures\\
  $(B, \land, \lor, \lnot, 0, 1)$ whose reducts $(B,\land,1)$ and $(B,\lor,0)$ are
  monoids, and that
  satisfy the equations $\lnot\lnot x =x$, and $\lnot(x\land y) =\lnot x\land\lnot
  y$, and also
\begin{align*}
  x\land y&=y\land x, & x\lor y&=y\lor x, \\
  x\land(y\lor z)&=(x\land y)\lor(x\land z),
         & x\lor(y\land z)&=(x\lor y)\land(x\lor z),\\
  x\land \lnot x&=0 & x\lor\lnot x&=1\\
  x\land 0 &=0 & x\lor 1&=1.
\end{align*}
  \item
  Find a structure $(C,+,\land,\curlyvee,\lnot,0,1)$ that satisfies
  \begin{equation*}
  x+y =(x\land\lnot y)\curlyvee(y\land\lnot x),
  \end{equation*}
  and whose reduct $(C,+,\land,0,1)$ is a Boolean ring, but whose reduct
  $(C,\land,\curlyvee,\lnot,0,1)$ is \emph{not} a Boolean algebra.
\end{enumerate}
\end{problem}
\begin{problem}
Prove that the map $x\mapsto[x]$ from a Boolean algebra to the power-set
of its Stone-space is an embedding of Boolean algebras.
\end{problem}
\end{document}
