Mathematical Structures: Monoidal t-norm logic algebras

# Monoidal t-norm logic algebras

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

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
 \Large Monoidal t-norm logic algebras \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Monoidal_t-norm_logic_algebras}{edit}

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

Changed: 35,40c34
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class} $op_1$ is (name of property): $axiom_1$
 A monoidal t-norm logic algebra is a \href{FLew-algebras.pdf}{FL$_{ew}$-algebra} $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \to, 0\rangle$ such that

Changed: 42c36
 $op_2$ is ...: $...$
 $\cdot$ is prelinear: $(x\to y)\vee (y\to x)=1$

Changed: 51,52c45,50
 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 monoidal t-norm logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$, $h(x \to y)=h(x) \to h(y)$, $h(0)=0$

Changed: 78c76
 Classtype & (value, see description) \cite{Ln19xx} \\\hline
 Classtype & variety \\\hline

Changed: 82,88c80,86
 Locally finite & \\\hline Residual size & \\\hline Congruence distributive & \\\hline Congruence modular & \\\hline Congruence $n$-permutable & \\\hline Congruence regular & \\\hline Congruence uniform & \\\hline
 Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & yes\\\hline Congruence modular & yes\\\hline Congruence $n$-permutable & yes, $n=2$\\\hline Congruence regular & no\\\hline Congruence uniform & no\\\hline

Changed: 120,122c118
 \href{....pdf}{...} subvariety \href{....pdf}{...} expansion
 \href{Basic_logic_algebras.pdf}{Basic logic algebras}

Changed: 128c124,128
 \href{....pdf}{...} supervariety
 \href{Representable_FLw_algebras.pdf}{Representable FL$_w$ algebras} \href{Representable_FLe_algebras.pdf}{Representable FL$_e$ algebras} \href{Distributive_FLew_algebras.pdf}{Distributive FL$_{ew}$ algebras}

Changed: 130c130
 \href{....pdf}{...} subreduct
 \href{Representable_commutative_integral_residuated_lattices.pdf}{Representable commutative integral residuated lattices} reduced type

http://mathcs.chapman.edu/structuresold/files/Monoidal_t-norm_logic_algebras.pdf
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\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
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\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
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\begin{document}
\textbf{\Large Monoidal t-norm logic algebras}

\abbreviation{MTLA}

\begin{definition}
A \emph{monoidal t-norm logic algebra} is a \href{FLew-algebras.pdf}{FL$_{ew}$-algebra} $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \to, 0\rangle$ such that

$\cdot$ is \emph{prelinear}:  $(x\to y)\vee (y\to x)=1$

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 monoidal t-norm logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x \vee y)=h(x) \vee h(y)$,
$h(x \wedge y)=h(x) \wedge h(y)$,
$h(x \cdot y)=h(x) \cdot h(y)$,
$h(x \to y)=h(x) \to h(y)$,
$h(0)=0$
\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               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & no\\\hline
Residual size                   & unbounded\\\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{Basic_logic_algebras.pdf}{Basic logic algebras}

\end{subclasses}

\begin{superclasses}\

\href{Representable_FLw_algebras.pdf}{Representable FL$_w$ algebras}

\href{Representable_FLe_algebras.pdf}{Representable FL$_e$ algebras}

\href{Distributive_FLew_algebras.pdf}{Distributive FL$_{ew}$ algebras}

\href{Representable_commutative_integral_residuated_lattices.pdf}{Representable commutative integral residuated lattices} reduced type

\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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