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\begin{document}
\textbf{\Large Abelian lattice-ordered groups}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Abelian_lattice-ordered_groups}{edit}
\abbreviation{AbLGrp}
\begin{definition}
An \emph{abelian lattice-ordered group} (or abelian $\ell $\emph{-group}) is a
\href{Lattice-ordered_group.pdf}{lattice-ordered group}
$\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ such that
$\cdot$ is commutative: $x\cdot y=y\cdot x$
\end{definition}
\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be $\ell$-groups. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $f:L\rightarrow M$ that is a
homomorphism: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\cdot y)=f(x)\cdot f(y)$.
Remark: It follows that $f(x\wedge y)=f(x)\wedge f(y)$, $f(x^{-1})=f(x)^{-1}$, and $f(e)=e$
\end{morphisms}
\begin{definition}
An \emph{abelian lattice-ordered group} (or \emph{abelian $\ell$-group}) is a
\href{Commutative_residuated_lattices.pdf}{commutative residuated lattice}
$\mathbf{L}=\langle L, \vee, \wedge, \cdot, \to, e\rangle $ that satisfies the identity
$x\cdot(x\to e)=e$.
Remark: $x^{-1}=x\to e$ and $x\to y=x^{-1}y$
\end{definition}
\begin{examples}
$\langle\mathbb{Z}, \mbox{max}, \mbox{min}, +, -, 0\rangle$, the integers with maximum, minimum, addition, unary subtraction and zero. The variety of abelian $\ell$-groups is generated by this algebra.
\end{examples}
\begin{basic_results}
The lattice reducts of (abelian) $\ell$-groups are \href{Distributive_lattices.pdf}{distributive lattices}.
\end{basic_results}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & hereditarily undecidable \cite{Gurevic1967} \cite{Burris1985}\\\hline
Locally finite & no\\\hline
Residual size & \\\hline
Congruence distributive & yes (see \href{Lattices.pdf}{lattices})\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$ (see \href{Groups.pdf}{groups})\\\hline
Congruence regular & yes, (see \href{Groups.pdf}{groups})\\\hline
Congruence uniform & yes, (see \href{Groups.pdf}{groups})\\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.
None
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\
\href{Totally_ordered_abelian_groups.pdf}{Totally ordered abelian groups}
\end{subclasses}
\begin{superclasses}\
\href{Representable_lattice-ordered_groups.pdf}{Representable lattice-ordered groups}
\end{superclasses}
\begin{thebibliography}{10}
\bibitem{Gurevic1967}
Yuri Gurevic, \emph{Hereditary undecidability of a class of lattice-ordered Abelian groups},
Algebra i Logika Sem.,
\textbf{6}, 1967, 45--62 \href{http://www.ams.org/mathscinet-getitem?mr=36:92}{MRreview}
\bibitem{Burris1985}
Stanley Burris, \emph{A simple proof of the hereditary undecidability of the theory of lattice-ordered abelian groups},
Algebra Universalis,
\textbf{20}, 1985, 400--401 \href{http://www.ams.org/mathscinet-getitem?mr=87g:06043}{MRreview}
\end{thebibliography}
\end{document}
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