Mathematical Structures: Intuitionistic linear logic algebras

# Intuitionistic linear logic algebras

Difference (from prior major revision) (no other diffs)

Changed: 34c34
 An intuitionistic linear logic algebra is a structure $\mathbf{A}=\langle A,\vee, \bottom, \wedge, \top, \cdot, 1, \backslash, /, 0, !\rangle$ of type $\langle 2, 0, 2, 0, 2, 0, 2, 2, 0, 1\rangle$ such that
 An intuitionistic linear logic algebra is a structure $\mathbf{A}=\langle A,\vee, \bot, \wedge, \top, \cdot, 1, \backslash, /, 0, !\rangle$ of type $\langle 2, 0, 2, 0, 2, 0, 2, 2, 0, 1\rangle$ such that

Changed: 38c38
 $\bottom$ is the least element: $\bottom\le x$
 $\bot$ is the least element: $\bot\le x$

http://mathcs.chapman.edu/structuresold/files/Intuitionistic_linear_logic_algebras.pdf
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\begin{document}
\textbf{\Large Intuitionistic linear logic algebras}

\abbreviation{ILLA}

\begin{definition}
An \emph{intuitionistic linear logic algebra} is a structure $\mathbf{A}=\langle A,\vee, \bot, \wedge, \top, \cdot, 1, \backslash, /, 0, !\rangle$ of type $\langle 2, 0, 2, 0, 2, 0, 2, 2, 0, 1\rangle$ such that

$\langle A,\vee, \wedge, \cdot, 1, \backslash, /, 0\rangle$ is a \href{FL-algebras.pdf}{FL-algebra}

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

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

$!$ is a closure operator:

$!$ satisfies ...:

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 ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x ... y)=h(x) ... h(y)$
\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                       & (value, see description) \cite{Ln19xx} \\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & \\\hline
Congruence modular              & \\\hline
Congruence $n$-permutable       & \\\hline
Congruence regular              & \\\hline
Congruence uniform              & \\\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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