# PlanetPhysics/Algebraic Category of LMn Logic Algebras

This is a topic entry on the algebraic category 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.

### Introduction

The \htmladdnormallink{category {http://planetphysics.us/encyclopedia/Cod.html} ${\displaystyle {\mathcal {LM}}}$ of \L{}ukasiewicz-Moisil, ${\displaystyle n}$-valued logic algebras (${\displaystyle LM_{n}}$), and ${\displaystyle LM_{n}}$--lattice morphisms}, ${\displaystyle \lambda _{LM_{n}}}$, was introduced in 1970 in ref. [1] as an algebraic category tool for ${\displaystyle n}$-valued logic studies. The objects of ${\displaystyle {\mathcal {LM}}}$ are the non--commutative ${\displaystyle LM_{n}}$ lattices and the morphisms of ${\displaystyle {\mathcal {LM}}}$ are the ${\displaystyle LM_{n}}$-lattice morphisms as defined here in the section following a brief historical note.

### History

\L{}ukasiewicz 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 (${\displaystyle LM_{n}}$) logic algebras were defined axiomatically in 1970, in ref. [1], as n-valued logic algebra representations and extensions of the \L ukasiewcz (3-valued) logics; then, the universal properties of categories of ${\displaystyle LM_{n}}$ -logic algebras were also investigated and reported in a series of recent publications ([2] and references cited therein). Recently, several modifications of ${\displaystyle LM_{n}}$-logic algebras are under consideration as valid candidates for representations of quantum logics, as well as for modeling non-linear biodynamics in genetic nets' or networks ([3]), and in single-cell organisms, or in tumor growth. For a recent review on ${\displaystyle n}$-valued logic algebras, and major published results, the reader is referred to [2].

### Definition of \L{

ukasiewicz--Moisil (LM), n-valued logic algebras}

\rm (reported by G. Moisil in 1941, cited in refs. [4]).

A {\it ${\displaystyle n}$--valued \L ukasiewicz--Moisil algebra}, ({\it ${\displaystyle LM_{n}}$--algebra}) is a structure of the form $\displaystyle (L,\vee,\wedge,N,(\phii)_{i\in\{1,\ldots,n-1\}},0,1)$ , subject to the following axioms:

• (L1) ${\displaystyle (L,\vee ,\wedge ,N,0,1)}$ is a {\it de Morgan algebra}, that is, a bounded distributive lattice with a decreasing involution ${\displaystyle N}$ satisfying the de Morgan property ${\displaystyle N({x\vee y})=Nx\wedge Ny}$;
• (L2) For each ${\displaystyle i\in \{1,\ldots ,n-1\}}$, $\displaystyle \phii:L\lra L$ is a lattice endomorphism;\footnote{ The $\displaystyle \phii$ 's are called the Chrysippian endomorphisms of ${\displaystyle L}$.}
• (L3) For each ${\displaystyle i\in \{1,\ldots ,n-1\},x\in L}$, $\displaystyle \phii(x)\vee N{\phii(x)}=1$ and $\displaystyle \phii(x)\wedge N{\phii(x)}=0$ ;
• (L4) For each ${\displaystyle i,j\in \{1,\ldots ,n-1\}}$, $\displaystyle \phii\circ\phi_{j}=\phi_{k}$ iff ${\displaystyle (i+j)=k}$;
• (L5) For each ${\displaystyle i,j\in \{1,\ldots ,n-1\}}$, ${\displaystyle i\leq j}$ implies $\displaystyle \phii\leq\phi_{j}$ ;
• (L6) For each ${\displaystyle i\in \{1,\ldots ,n-1\}}$ and ${\displaystyle x\in L}$, $\displaystyle \phii(N x)=N\phi_{n-i}(x)$ .
• (L7) Moisil's determination principle': $\displaystyle \left[\orc i\in\{1,\ldots,n-1\},\;\phii(x)=\phii(y)\right] \; implies \; [x = y] \;.$

\begin{exe}\rm Let ${\displaystyle L_{n}=\{0,1/(n-1),\ldots ,(n-2)/(n-1),1\}}$. This set can be naturally endowed with an ${\displaystyle {\mbox{LM}}_{n}}$ --algebra structure as follows:

• the bounded lattice operations are those induced by the usual order on rational numbers;
• for each ${\displaystyle j\in \{0,\ldots ,n-1\}}$, ${\displaystyle N(j/(n-1))=(n-j)/(n-1)}$;
• for each ${\displaystyle i\in \{1,\ldots ,n-1\}}$ and ${\displaystyle j\in \{0,\ldots ,n-1\}}$, $\displaystyle \phii(j/(n-1))=0$ if ${\displaystyle j and ${\displaystyle =1}$ otherwise.

\end{exe} Note that, for ${\displaystyle n=2}$, ${\displaystyle L_{n}=\{0,1\}}$, and there is only one Chrysippian endomorphism of ${\displaystyle L_{n}}$ is ${\displaystyle \phi _{1}}$, which is necessarily restricted by the determination principle to a bijection, thus making ${\displaystyle L_{n}}$ a Boolean algebra (if we were also to disregard the redundant bijection ${\displaystyle \phi _{1}}$). Hence, the overloaded' notation ${\displaystyle L_{2}}$, which is used for both the classical Boolean algebra and the two--element ${\displaystyle {\mbox{LM}}_{2}}$--algebra, remains consistent. \begin{exe}\rm Consider a Boolean algebra $\displaystyle (B,\v,\w,{}^-,0,1)$ . Let ${\displaystyle T(B)=\{(x_{1},\ldots ,x_{n})\in B^{n-1}\mid x_{1}\leq \ldots \leq x_{n-1}\}$.Ontheset}$T(B)${\displaystyle ,wedefinean}$\mbox{LM}_n$-algebra structure as follows:

• the lattice operations, as well as ${\displaystyle 0}$ and ${\displaystyle 1}$, are defined component--wise from $\displaystyle \Ld$ ;
• for each ${\displaystyle (x_{1},\ldots ,x_{n-1})\in T(B)}$ and ${\displaystyle i\in \{1,\ldots ,n-1\}}$ one has:\\ $\displaystyle N(x_1,\ldots x_{n-1})=(\ov{x_{n-1}},\ldots,\ov{x_1})$ and $\displaystyle \phii(x_1,\ldots,x_n)=(x_i,\ldots,x_i) .$

\end{exe}

## References

1. Georgescu, G. and C. Vraciu. 1970, On the characterization of centered \L{}ukasiewicz algebras., J. Algebra , 16 : 486-495.
2. Georgescu, G. 2006, N-valued Logics and \L ukasiewicz-Moisil Algebras, Axiomathes , 16 (1-2): 123-136.
3. Baianu, I.C.: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology , 39 : 249-258.
4. Cite error: Invalid <ref> tag; no text was provided for refs named GG-CV70,GG2k6
5. Georgescu, G. and D. Popescu. 1968, On Algebraic Categories, Revue Roumaine de Math\'ematiques Pures et Appliqu\'ees , 13 : 337-342.