Mathematical Structures: Wajsberg hoops

# Wajsberg hoops

Difference (from prior major revision) (author diff)

Changed: 31c31
 A Wajsberg hoop is a \href{Hoops.pdf}{hoops} $\mathbf{A}=\langle A, \cdot, \rightarrow, 1\rangle$ such that
 A Wajsberg hoop is a \href{Hoops.pdf}{hoop} $\mathbf{A}=\langle A, \cdot, \rightarrow, 1\rangle$ such that

Removed: 35d34

Removed: 37d35

Changed: 65c65
 Congruence regular & yes Radim Belohlávek,On the regularity of MV-algebras and Wajsberg hoops,
 Congruence regular & yes Radim Belohlovek, On the regularity of MV-algebras and Wajsberg hoops,

Changed: 67c67
 442000,375--377\href{"http://www.ams.org/mathscinet-getitem?mr=1 816 031:"}{MRreview}\\\hline
 44, 2000, 375--377\href{http://www.ams.org/mathscinet-getitem?mr=1 816 031}{MRreview}\\\hline

http://mathcs.chapman.edu/structuresold/files/Wajsberg_hoops.pdf
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\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Wajsberg hoops}

\begin{definition}
A \emph{Wajsberg hoop} is a \href{Hoops.pdf}{hoop} $\mathbf{A}=\langle A, \cdot, \rightarrow, 1\rangle$ such that

$(x\rightarrow y)\rightarrow y = (y\rightarrow x)\rightarrow x$

Remark: Lattice operations are term-definable by $x\wedge y=x\cdot(x\rightarrow y)$ and $x\vee y=(x\rightarrow y)\rightarrow y$.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be Wajsberg hoops. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$, $h(1)=1$
\end{morphisms}

\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})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable\\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & no\\\hline
Residual size & \\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & \\\hline
Congruence regular & yes Radim Belohlovek, \emph{On the regularity of MV-algebras and Wajsberg hoops},
Algebra Universalis,
\textbf{44}, 2000, 375--377\href{http://www.ams.org/mathscinet-getitem?mr=1 816 031}{MRreview}\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Generalized_Boolean_algebras.pdf}{Generalized Boolean algebras}

\href{MV-algebras.pdf}{MV-algebras}

\end{subclasses}
\begin{superclasses}\

\href{Hoops.pdf}{Hoops}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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