Mathematical Structures: Lattice-ordered groups

# Lattice-ordered groups

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http://mathcs.chapman.edu/structuresold/files/Lattice-ordered_groups.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Lattice-ordered groups}

\begin{definition}
A \emph{lattice-ordered group} (or $\ell$\emph{-group}) is a structure $\mathbf{L}=\langle L, \vee, \wedge, \cdot, ^{-1}, e\rangle$ such that

$\langle L, \vee, \wedge\rangle$ is a \href{Lattices.pdf}{lattice}

$\langle L, \cdot, ^{-1}, e\rangle$ is a \href{Groups.pdf}{group}

$\cdot$ is order-preserving:  $x\leq y\implies uxv\leq uyv$

Remark:
$xy=x\cdot y$, $x\leq y\Longleftrightarrow x\wedge y=x$ and $x\leq y\Longleftrightarrow x\vee y=y$

\end{definition}
\begin{definition}
A \emph{lattice-ordered group} (or $\ell$\emph{-group}) is a structure $\mathbf{L}=\left\langle L,\vee ,\cdot ,^{-1},e\right\rangle$ such that

$\langle L,\vee\rangle$ is a \href{Semilattices.pdf}{semilattice}

$\langle L,\cdot,^{-1},e\rangle$ is a \href{Groups.pdf}{group}

$\cdot$ is join-preserving:  $u(x\vee y)v=uxv\vee uyv$

Remark: $x\wedge y=\left( x^{-1}\vee y^{-1}\right) ^{-1}$
\end{definition}

\begin{definition}
A \emph{lattice-ordered group} (or $\ell$\emph{-group}) is a residuated
lattice $\mathbf{L}=\langle L,\vee ,\wedge ,\cdot ,\backslash ,/,e\rangle$ that satisfies the identity $x(e/x)=e$.

Remark: $x^{-1}=e/x=x\backslash e$, $x/y=xy^{-1}$ and $x\backslash y=x^{-1}y$
\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: $f(x\vee y)=f(x)\vee f(y)$, $f(x\wedge y)=f(x)\wedge f(y)$, $f(x\cdot y)=f(x)\cdot f(y)$, $f(x^{-1})=f(x)^{-1}$, and $f(e)=e$.
\end{morphisms}
\begin{basic_results}
The lattice reducts of lattice-ordered groups are \href{Distributive_lattices.pdf}{distributive lattices}.
\end{basic_results}
\begin{examples}
$\langle Aut(\mathbf{C}),\mbox{max},\mbox{min},\circ,^{-1},id_{\mathbf{C}}\rangle$,
the group of order-automorphisms of a \href{Chains.pdf}{Chains} $\mathbf{C}$, with $\mbox{max}$ and $\mbox{min}$
(applied pointwise), composition, inverse, and identity automorphism.
\end{examples}
\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

W. Charles Holland,Stephen H. McCleary,\emph{Solvability of the word problem in free lattice-ordered groups},
Houston J. Math.,
\textbf{5}1979,99--105\href{"http://www.ams.org/mathscinet-getitem?mr=80f:06018"}{MRreview}\href{"http://www.emis.de/MATH-item?0404.06009"}{ZMATH}
[http://www.chapman.edu/~jipsen/lgroups/lgroupDecisionProc.html implementation]\\\hline

Quasiequational theory & undecidable

A. M. W. Glass,Yuri Gurevich,\emph{The word problem for lattice-ordered groups},
Trans. Amer. Math. Soc.,
\textbf{280}1983,127--138\href{"http://www.ams.org/mathscinet-getitem?mr=85d:06015"}{MRreview}\href{"http://www.emis.de/MATH-item?0586.03037"}{ZMATH}\\\hline

First-order theory & hereditarily undecidable

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}\href{"http://www.emis.de/MATH-item?0165.31803"}{ZMATH}

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}\href{"http://www.emis.de/MATH-item?0575.06017"}{ZMATH}\\\hline

Congruence distributive & yes, see \href{Lattices.pdf}{lattices}\\\hline
Congruence extension property & \\\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

Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & no

Keith R. Pierce,\emph{Amalgamations of lattice ordered groups},
Trans. Amer. Math. Soc.,
\textbf{172}1972,249--260\href{"http://www.ams.org/mathscinet-getitem?mr=48 :3835"}{MRreview}\href{"http://www.emis.de/MATH-item?0259.06017"}{ZMATH}\\\hline
Strong amalgamation property & no\\\hline
Epimorphisms are surjective & \\\hline

\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} None \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\

\href{Normal_valued_lattice_ordered_groups.pdf}{Normal valued lattice ordered groups}

\end{subclasses}
\begin{superclasses}\

\href{Cancellative_residuated_lattices.pdf}{Cancellative residuated lattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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