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 catgory ${\mathcal {LM}}$ of Łukasiewicz–Moisil, $n$-valued logic algebras ($LM_{n}$), and $LM_{n}$–lattice morphisms}, $\lambda _{LM_{n}}$, was introduced in 1970 in ref. as an algebraic category tool for $n$-valued logic studies. The objects of ${\mathcal {LM}}$ are the non-commutative $LM_{n}$ lattices and the morphisms of ${\mathcal {LM}}$ are the $LM_{n}$-lattice morphisms as defined here in the section following a brief historical note.

History

Łukasiewicz 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. Łukasiewicz–Moisil ($LM_{n}$) logic algebras were defined axiomatically in 1970, in ref. , as n-valued logic algebra representations 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 ( and references cited therein). Recently, several modifications of $LM_{n}$-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 (), 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 .

Definition of LM_n

(reported by G. Moisil in 1941, cited in refs. ).

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

• (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 $N({x\vee y})=Nx\wedge Ny$;
• (L2) For each $i\in \{1,\ldots ,n-1\}$, $\phi _{i}:L\leftrightarrow L$ is a lattice endomorphism. (The $\phi$'s are called the Chrysippian endomorphisms of $L$.)
• (L3) For each $i\in \{1,\ldots ,n-1\},x\in L$, $\phi _{i}(x)\vee N{\phi _{i}(x)}=1$ and $\phi (x)\wedge N{\phi (x)}=0$;
• (L4) For each $i,j\in \{1,\ldots ,n-1\}$, $\phi _{i}\circ \phi _{j}=\phi _{k}$ iff $(i+j)=k$;
• (L5) For each $i,j\in \{1,\ldots ,n-1\}$, $i\leq j$ implies $\phi _{i}\leq \phi _{j}$;
• (L6) For each $i\in \{1,\ldots ,n-1\}$ and $x\in L$, $\phi (Nx)=N\phi _{n-i}(x)$.
• (L7) Moisil's determination principle: Failed to parse (syntax error): {\displaystyle \text{For} i\in\{1,\ldots,n-1\},\;\phi_i(x)=\phi_i(y)\right] \; implies \; [x = y] \;.}

\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:

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

\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 $T(B)=\{(x_{1},\ldots ,x_{n})\in B^{n-1}\mid x_{1}\leq \ldots \leq x_{n-1}\}<math>.Ontheset$T(B)$,wedefinean$\mbox{LM}_n</math>-algebra structure as follows:

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

\end{exe}