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%%% Primary Title: algebraic category of LMn -logic algebras
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\begin{document}
This is a topic entry on the \htmladdnormallink{algebraic category}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} of \L{}ukasiewicz--Moisil n-valued logic algebras that provides basic concepts and the background of the modern development in this area of many-valued logics.
\subsection{Introduction}
The \emph{\htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{LM}$ of \L{}ukasiewicz-Moisil, $n$-valued logic algebras ($LM_n$), and $LM_n$--lattice morphisms}, $\lambda_{LM_n}$, was introduced in 1970 in ref. \cite{GG-CV70} as an algebraic category tool for $n$-valued logic studies. The \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $\mathcal{LM}$ are the \emph{non--commutative} $LM_n$ lattices and the \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of $\mathcal{LM}$ are the $LM_n$-lattice morphisms as defined here in the \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} following a brief historical note.
\subsection{History}
\L{}ukasiewicz \emph{logic algebras} were constructed by Grigore Moisil in 1941 to define `nuances' in logics, or many-valued logics, as well as 3-state control logic (electronic) circuits. \L{}ukasiewicz-Moisil ($LM_n$) logic algebras were defined axiomatically in 1970, in ref. \cite{GG-CV70}, as n-valued logic algebra \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} and extensions of the \L ukasiewcz (3-valued) logics; then, the universal properties of categories of $LM_n$ -logic algebras were also investigated and reported in a series of recent publications (\cite{GG2k6} and references cited therein).
Recently, several modifications of {\em $LM_n$-logic algebras} are under consideration as valid candidates for representations of {\em \htmladdnormallink{quantum logics}{http://planetphysics.us/encyclopedia/LQG2.html}}, as well as for modeling non-linear biodynamics in genetic `nets' or networks (\cite{ICB77}), and in single-cell organisms, or in tumor growth. For a recent review on $n$-valued logic algebras, and major published results, the reader is referred to \cite{GG2k6}.
\subsection{Definition of \L{}ukasiewicz--Moisil (LM), n-valued logic algebras}
\begin{definition}\rm (reported by G. Moisil in 1941, cited in refs. \cite{GG-CV70,GG2k6}).
A {\it $n$--valued \L ukasiewicz--Moisil algebra}, ({\it $LM_{n}$--algebra}) is a structure of the form
$(L,\vee,\wedge,N,(\phii)_{i\in\{1,\ldots,n-1\}},0,1)$, subject to the following axioms:
\begin{itemize}
\item (L1) $(L,\vee,\wedge,N,0,1)$ is a {\it de Morgan algebra}, that is, a bounded distributive lattice with a decreasing involution $N$ satisfying the de Morgan property $N({x\vee y})=Nx\wedge Ny$;
\item (L2) For each $i\in\{1,\ldots,n-1\}$, $\phii:L\lra L$ is a lattice endomorphism;\footnote{ The $\phii$'s are called the {\em Chrysippian endomorphisms} of $L$.}
\item (L3) For each $i\in\{1,\ldots,n-1\},x\in L$, $\phii(x)\vee N{\phii(x)}=1$ and
$\phii(x)\wedge N{\phii(x)}=0$;
\item (L4) For each $i,j\in\{1,\ldots,n-1\}$, $\phii\circ\phi_{j}=\phi_{k}$ iff $(i+j)= k$;
\item (L5) For each $i,j\in\{1,\ldots,n-1\}$, $i\leq j$ implies $\phii\leq\phi_{j}$;
\item (L6) For each $i\in\{1,\ldots,n-1\}$ and $x\in L$, $\phii(N x)=N\phi_{n-i}(x)$.
\item (L7) Moisil's `determination principle':
$$\left[\orc i\in\{1,\ldots,n-1\},\;\phii(x)=\phii(y)\right] \; implies \; [x = y] \;.$$
\end{itemize}
\end{definition}
\begin{exe}\rm
Let $L_n=\{0,1/(n-1),\ldots,(n-2)/(n-1),1\}$. This set can be naturally endowed with an $\mbox{LM}_n$
--algebra structure as follows:
\begin{itemize}
\item the bounded lattice \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} are those induced by the usual order on rational numbers;
\item for each $j\in\{0,\ldots,n-1\}$, $N(j/(n-1))=(n-j)/(n-1)$;
\item for each $i\in\{1,\ldots,n-1\}$ and $j\in\{0,\ldots,n-1\}$,
$\phii(j/(n-1))=0$ if $j<i$ and $=1$ otherwise.
\end{itemize}
\end{exe}
Note that, for $n=2$, $L_n=\{0,1\}$, and there is only one Chrysippian endomorphism of $L_n$ is $\phi_1$, which
is necessarily restricted by the determination principle to a bijection, thus making $L_n$ a Boolean algebra (if
we were also to disregard the redundant bijection $\phi_1$). Hence, the `overloaded' notation $L_2$, which is
used for both the classical Boolean algebra and the two--element $\mbox{LM}_2$--algebra, remains consistent.
\begin{exe}\rm
Consider a Boolean algebra $(B,\v,\w,{}^-,0,1)$. Let $T(B)=\{(x_1,\ldots,x_n)\in B^{n-1}\mid x_1\leq\ldots\leq
x_{n-1}\}$. On the set $T(B)$, we define an $\mbox{LM}_n$-algebra structure as follows:
\begin{itemize}
\item the lattice operations, as well as $0$ and $1$, are defined component--wise from $\Ld$;
\item for each $(x_1,\ldots,x_{n-1})\in T(B)$ and $i\in\{1,\ldots,n-1\}$ one has:\\
$N(x_1,\ldots x_{n-1})=(\ov{x_{n-1}},\ldots,\ov{x_1})$ and $\phii(x_1,\ldots,x_n)=(x_i,\ldots,x_i) .$
\end{itemize}
\end{exe}
\begin{thebibliography}{9}
\bibitem{GG-CV70}
Georgescu, G. and C. Vraciu. 1970, On the characterization of centered \L{}ukasiewicz
algebras., {\em J. Algebra}, \textbf{16}: 486-495.
\bibitem{GG2k6}
Georgescu, G. 2006, N-valued Logics and \L ukasiewicz-Moisil
Algebras, \emph{Axiomathes}, \textbf{16} (1-2): 123-136.
\bibitem{ICB77}
Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. \emph{Bulletin of Mathematical Biology}, \textbf{39}: 249-258.
\bibitem{GGDP68}
Georgescu, G. and D. Popescu. 1968, On Algebraic Categories,
\emph{Revue Roumaine de Math\'ematiques Pures et Appliqu\'ees}, \textbf{13}:
337-342.
\end{thebibliography}
\end{document}
