Mathematical Structures: Multiplicative additive linear logic algebras

# Multiplicative additive linear logic algebras

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Changed: 28,30c28,29
 \Large Name of class \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Template}{edit} % Note: replace "Template" with Name_of_class in previous line

Changed: 32c31
 \abbreviation{Abbr}
 \abbreviation{MALLA}

Changed: 35,36c34,39
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that
 A multiplicative additive linear logic algebra is a structure $\mathbf{A}=\langle A,\vee,\bot,\wedge,\top,+,0,\cdot,1,^\perp\rangle$ of type $\langle 2,0,2,0,2,0,1\rangle$ such that $\langle A,\vee,\wedge,\cdot,1,^{\perp}\rangle$ is a \href{Commutative_involutive_residuated_lattices.pdf}{commutative involutive residuated lattice} $\bot$ is the least element: $\bot\le x$

Changed: 38c41
 $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 $\top$ is the greatest element: $x\le\top$

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 $op_1$ is (name of property): $axiom_1$
 $+$ is the dual of $\cdot$: $x+y=(x^\perp\cdot y^\perp)^\perp$

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 $op_2$ is ...: $...$
 $0$ is the dual of $1$: $0=1^\perp$

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 Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$
 Let $\mathbf{A}$ and $\mathbf{B}$ be multiplicative additive linear logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism.

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 Classtype & (value, see description) \cite{Ln19xx} \\\hline Equational theory & \\\hline
 Classtype & variety \\\hline Equational theory & decidable\\\hline

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 Congruence distributive & \\\hline Congruence modular & \\\hline Congruence $n$-permutable & \\\hline Congruence regular & \\\hline Congruence uniform & \\\hline
 Congruence distributive & yes\\\hline Congruence modular & yes\\\hline Congruence $n$-permutable & yes, $n=2$\\\hline Congruence regular & no\\\hline Congruence uniform & no\\\hline

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\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
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\begin{document}
\textbf{\Large Multiplicative additive linear logic algebras}

\abbreviation{MALLA}

\begin{definition}
A \emph{multiplicative additive linear logic algebra} is a structure $\mathbf{A}=\langle A,\vee,\bot,\wedge,\top,+,0,\cdot,1,^\perp\rangle$ of type $\langle 2,0,2,0,2,0,1\rangle$ such that

$\langle A,\vee,\wedge,\cdot,1,^{\perp}\rangle$ is a \href{Commutative_involutive_residuated_lattices.pdf}{commutative involutive residuated lattice}

$\bot$ is the least element:  $\bot\le x$

$\top$ is the greatest element:  $x\le\top$

$+$ is the dual of $\cdot$: $x+y=(x^\perp\cdot y^\perp)^\perp$

$0$ is the dual of $1$: $0=1^\perp$

Remark: This is a template.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be multiplicative additive linear logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism.
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is ...:  $axiom$

$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
Classtype                       & variety \\\hline
Equational theory               & decidable\\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & yes\\\hline
Congruence modular              & yes\\\hline
Congruence $n$-permutable       & yes, $n=2$\\\hline
Congruence regular              & no\\\hline
Congruence uniform              & no\\\hline
Congruence extension property   & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong.     & \\\hline
Amalgamation property           & \\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

\end{finite_members}

\begin{subclasses}\

\href{....pdf}{...} subvariety

\href{....pdf}{...} expansion

\end{subclasses}

\begin{superclasses}\

\href{....pdf}{...} supervariety

\href{....pdf}{...} subreduct

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}

\end{thebibliography}

\end{document}
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