MV-algebras

http://mathcs.chapman.edu/structuresold/files/MV-algebras.pdf

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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large MV-algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=MV-algebras}{edit}

\abbreviation{MV}

\begin{definition}
An \emph{MV-algebra} (short for \emph{multivalued logic algebra}) is a
structure $\mathbf{A}=\langle A, +, 0, \neg\rangle$ such that


$\langle A, +, 0\rangle$ is a \href{Commutative_monoids.pdf}{commutative monoid}


$\neg \neg x=x$


$x + \neg 0 = \neg 0$


$\neg(\neg x+y)+y = \neg(\neg y+x)+x$

Remark: This is the definition from \cite{COM2000}

\end{definition}

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

$h(x+y)=h(x)+h(y)$, $h(\neg x)=\neg h(x)$, $h(0)=0$
\end{morphisms}

\begin{definition}
An \emph{MV-algebra} is a
structure $\mathbf{A}=\langle A, +, 0, \cdot, 1, \neg\rangle$ such that


$\langle A, \cdot, 1\rangle$ is a \href{Commutative_monoids.pdf}{commutative monoid}


$\neg $ is a DeMorgan involution for $+,\cdot $:  $\neg \neg x=x$, $x+y=\neg \left( \neg x\cdot \neg
y\right) $


$\neg 0=1$, $0\cdot x=0$, $\neg \left( \neg
x+y\right) +y=\neg \left( \neg y+x\right) +x$
\end{definition}

\begin{definition}
An \emph{MV-algebra} is a \href{Basic_logic_algebras.pdf}{basic logic algebra} $\mathbf{A}=\langle
A,\vee,0,\wedge,1,\cdot,\rightarrow\rangle$ that satisfies

MV:  $x\vee y=(x\rightarrow y)\rightarrow y$
\end{definition}

\begin{definition}
A \emph{Wajsberg algebra} is an algebra $\mathbf{A}=\langle A, \rightarrow, \neg, 1\rangle$ such that


$1\rightarrow x=x$


$(x\rightarrow y)\rightarrow ((y\rightarrow z) \rightarrow (x\rightarrow z) = 1$


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


$(\neg x\rightarrow \neg y)\rightarrow(y\rightarrow x)=1$

Remark: 
Wajsberg algebras are term-equivalent to MV-algebras via $x\rightarrow y=\neg x+y$, $1=\neg 0$ and $x + y=\neg x\rightarrow y$, $0=\neg 1$.

\end{definition}

\begin{definition}
A \emph{bounded hoop} is an algebra $\mathbf{A}=\langle A, \cdot, \rightarrow, 0, 1\rangle$ such that

$\langle A, \cdot, \rightarrow, 1\rangle$ is a hoop

$0\rightarrow x=1$

Remark: 
Bounded hoops are term-equivalent to Wajsberg algebras via $x\cdot y=\neg(x\rightarrow\neg y)$, $0=\neg 1$, and $\neg x=x\rightarrow 0$.
See \cite{BP1994} for details.

\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})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable\\\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 e-regular & yes, $e=1$\\\hline
Congruence uniform & \\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & yes \cite{Mu1987}\\\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 $n$.

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

\href{Boolean_algebras.pdf}{Boolean algebras} 

\end{subclasses}
\begin{superclasses}\ 

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

\href{Basic_logic_algebras.pdf}{Basic logic algebras} 

\href{Wajsberg_hoops.pdf}{Wajsberg hoops} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{BP1994}
W. J. Blok, D. Pigozzi,
\emph{On the structure of varieties with equationally definable principal congruences. III},
Algebra Universalis,
\textbf{32} 1994, 545--608 \href{http://www.ams.org/mathscinet-getitem?mr=96b:08007}{MRreview}

\bibitem{COM2000}
Roberto L. O. Cignoli, Itala M. L. D'Ottaviano, Daniele Mundici,
\emph{Algebraic foundations of many-valued reasoning},
Trends in Logic---Studia Logica Library
\textbf{7} Kluwer Academic Publishers
2000, x+231 \href{http://www.ams.org/mathscinet-getitem?mr=2001j:03114}{MRreview}

\bibitem{Mu1987}
Daniele Mundici,
\emph{Bounded commutative BCK-algebras have the amalgamation property},
Math. Japon.,
\textbf{32} 1987, 279--282 
\href{http://www.ams.org/mathscinet-getitem?mr=88i:06020}{MRreview}

\end{thebibliography}

\end{document}
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